The Tits Alternative for Out(Fn) II: a Kolchin Type Theorem

Annals of Mathematics, 161 (2005), 1–59

The Tits alternative for Out(Fn) II: A Kolchin type theorem

By Mladen Bestvina, Mark Feighn, and Michael Handel*

Abstract

This is the second of two papers in which we prove the Tits alternative for Out(Fn).

Contents 1. Introduction and outline

2. Fn-trees 2.1. Real trees 2.2. Real Fn-trees 2.3. Very small trees 2.4. Spaces of real Fn-trees 2.5. Bounded cancellation constants 2.6. Real graphs 2.7. Models and normal forms for simplicial Fn-trees 2.8. Free factor systems 3. Unipotent polynomially growing outer automorphisms 3.1. Unipotent linear maps 3.2. Topological representatives 3.3. Relative train tracks and automorphisms of polynomial growth 3.4. Unipotent representatives and UPG automorphisms 4. The dynamics of unipotent automorphisms 4.1. Polynomial sequences 4.2. Explicit limits 4.3. Primitive subgroups 4.4. Unipotent automorphisms and trees 5. A Kolchin theorem for unipotent automorphisms 5.1. F contains the suﬃxes of all nonlinear edges 5.2. Bouncing sequences stop growing 5.3. Bouncing sequences never grow

*The authors gratefully acknowledge the support of the National Science Foundation. 2 MLADEN BESTVINA, MARK FEIGHN, AND MICHAEL HANDEL

5.4. Finding Nielsen pairs 5.5. Distances between the vertices 5.6. Proof of Theorem 5.1 6. Proof of the main theorem References

1. Introduction and outline

Recent years have seen a development of the theory for Out(Fn), the outer automorphism group of the free group Fn of rank n, that is modeled on Nielsen- Thurston theory for surface homeomorphisms. As mapping classes have either exponential or linear growth rates, so free group outer automorphisms have either exponential or polynomial growth rates. (The degree of the polynomial can be any integer between 1 and n−1; see [BH92].) In [BFH00], we considered individual automorphisms with primary emphasis on those with exponential growth rates. In this paper, we focus on subgroups of Out(Fn) all of whose elements have polynomial growth rates. To remove certain technicalities arising from finite order phenomena, we restrict our attention to those outer automorphisms of polynomial growth ∼ n whose induced automorphism of H1(Fn; Z) = Z is unipotent. We say that such an outer automorphism is unipotent. The subset of unipotent outer automorphisms of Fn is denoted UPG(Fn) (or just UPG). A subgroup of Out(Fn) is unipotent if each element is unipotent. We prove (Proposition 3.5) that any polynomially growing outer automorphism that acts trivially in Z/3Z- homology is unipotent. Thus every subgroup of polynomially growing outer automorphisms has a finite index unipotent subgroup. The archetype for the main theorem of this paper comes from linear groups. A linear map is unipotent if and only if it has a basis with respect to which it is upper triangular with 1’s on the diagonal. A celebrated theorem of Kolchin [Ser92] states that for any group of unipotent linear maps there is a basis with respect to which all elements of the group are upper triangular with 1’s on the diagonal. There is an analogous result for mapping class groups. We say that a mapping class is unipotent if it has linear growth and if the induced linear map on first homology is unipotent. The Thurston classification theorem implies that a mapping class is unipotent if and only if it is represented by a composition of Dehn twists in disjoint simple closed curves. Moreover, if a pair of unipotent mapping classes belongs to a unipotent subgroup, then their twisting curves cannot have transverse intersections (see for example [BLM83]). Thus every unipotent mapping class subgroup has a characteristic set of disjoint simple closed curves and each element of the subgroup is a composition of Dehn twists along these curves. THE TITS ALTERNATIVE FOR Out(Fn) II 3

Our main theorem is the analogue of Kolchin’s theorem for Out(Fn). Fix once-and-for-all a wedge Rosen of n circles and permanently identify its fundamental group with Fn. A marked graph (of rank n) is a graph equipped with a homotopy equivalence from Rosen; see [CV86]. A homotopy equivalence f : G → G on a marked graph G induces an outer automorphism of the fundamental group of G and therefore an element O of Out(Fn); we say that f : G → G is a representative of O. Suppose that G is a marked graph and that ∅ = G0 G1 ··· GK = G is a filtration of G where Gi is obtained from Gi−1 by adding a single edge Ei. A homotopy equivalence f : G → G is upper triangular with respect to the filtration if each f(Ei)=viEiui (as edge paths) where ui and vi are closed paths in Gi−1. If the choice of filtration is clear then we simply say that f : G → G is upper triangular. We refer to the ui’s and vi’s as suffixes and prefixes respectively. An outer automorphism is unipotent if and only if it has a representative that is upper triangular with respect to some filtered marked graph G (see Section 3). For any filtered marked graph G, let Q be the set of upper triangular homotopy equivalences of G up to homotopy relative to the vertices of G.By Lemma 6.1, Q is a group under the operation induced by composition. There is a natural map from Q to UPG(Fn). We say that a unipotent subgroup of Out(Fn)isfiltered if it lifts to a subgroup of Q for some filtered marked graph. We denote the conjugacy class of a free factor F i by [[F i]]. If F 1 ∗F 2 ∗···∗ F k is a free factor, then we say that the collection F = {[[F 1]], [[F 2]],...,[[F k]]} is a free factor system. There is a natural action of Out(Fn) on free factor systems and we say that F is H-invariant if each element of the subgroup H fixes F. A (not necessarily connected) subgraph K of a marked real graph determines a free factor system F(K). A partial order on free factor systems is defined in subsection 2.8. We can now state our main theorem.

Theorem 1.1 (Kolchin theorem for Out(Fn)). Every finitely generated unipotent subgroup H of Out(Fn) is filtered. For any H-invariant free factor system F, the marked filtered graph G can be chosen so that F(Gr)=F for some filtration element Gr. The number of edges of G can be taken to be 3n − bounded by 2 1 for n>1.

It is an interesting question whether or not the requirement that H be ﬁnitely generated is necessary or just an artifact of our proof.

Question. Is every unipotent subgroup of Out(Fn) contained in a ﬁnitely generated unipotent subgroup? 4 MLADEN BESTVINA, MARK FEIGHN, AND MICHAEL HANDEL

Remark 1.2. In contrast to unipotent mapping class subgroups which are all ﬁnitely generated and abelian, unipotent subgroups of Out(Fn) can be quite large. For example, if G is a wedge of n circles, then a ﬁltration on G corresponds to an ordered basis {e1,...,en} of Fn and elements of Q correspond to automorphisms of the form ei → aieibi with ai,bi ∈e1,...,ei−1. When n>2, the image of Q in UPG(Fn) contains a product of nonabelian free groups. This is the second of two papers in which we establish the Tits alternative for Out(Fn).

Theorem (The Tits alternative for Out(Fn)). Let H be any subgroup of Out(Fn). Then either H is virtually solvable, or contains a nonabelian free group. For a proof of a special (generic) case, see [BFH97a]. The following corollary of Theorem 1.1 gives another special case of the Tits alternative for Out(Fn). The corollary is then used to prove the full Tits alternative.

Corollary 1.3. Every unipotent subgroup H of Out(Fn) either contains a nonabelian free group or is solvable.

Proof. We first prove that if Q is defined as above with respect to a marked filtered graph G, then every subgroup Z of Q either contains a nonabelian free group or is solvable. Let i ≥ 0 be the largest parameter value for which every element of Z restricts to the identity on Gi−1.Ifi = K + 1, then Z is the trivial group and we are done. Suppose then that i ≤ K. By construction, each element of Z satisfies Ei → viEiui where vi and ui are paths (that depend on the element of Z)inGi−1 and are therefore fixed by every element of Z. The suffix map S : Z→Fn, which assigns the suffix ui to the element of Z, is therefore a homomorphism. The prefix map P : Z→Fn, which assigns the inverse of vi to the element of Z, is also a homomorphism. If the image of P×S: Z→Fn × Fn contains a nonabelian free group, then so does Z and we are done. If the image of P×Sis abelian then, since Z is an abelian extension of the kernel of P×S, it suffices to show that the kernel of P×Sis either solvable or contains a nonabelian free group. Upward induction on i now completes the proof. In fact, this argument shows that Z 3n − is polycyclic and that the length of the derived series is bounded by 2 1 for n>1. For H finitely generated the corollary now follows from Theorem 1.1. When H is not finitely generated, it can be represented as the increasing union of finitely generated subgroups. If one of these subgroups contains a nonabelian free group, then so does H, and if not then H is solvable with the 3n − length of the derived series bounded by 2 1. THE TITS ALTERNATIVE FOR Out(Fn) II 5

Proof of the Tits alternative for Out(Fn). Theorem 7.0.1 of [BFH00] asserts that if H does not contain a nonabelian free group then there is a ﬁnite index subgroup H0 of H and an exact sequence

1 →H1 →H0 →A→1 with A a ﬁnitely generated free abelian group and with H1 a unipotent subgroup of Out(Fn). Since H1 does not contain a nonabelian free group, by Corollary 1.3, H1 is solvable. Thus, H0 is solvable and H is virtually solvable.

In [BFH04] we strengthen the Tits alternative for Out(Fn) further by proving:

Theorem (Solvable implies abelian). A solvable subgroup of Out(Fn) has a ﬁnitely generated free abelian subgroup of index at most 35n2 .

Emina Alibegovi´c [Ali02] has since provided an alternate shorter proof. The rank of an abelian subgroup of Out(Fn)is≤ 2n − 3 for n>1 [CV86]. We reformulate Theorem 1.1 in terms of trees, and it is in this form that we prove the theorem. There is a natural right action of the automorphism group of Fn on the set of simplicial Fn-trees produced by twisting the action. See Section 2 for details. If we identify trees that are equivariantly isomorphic then this action descends to give an action of Out(Fn). A simplicial Fn-tree is nontrivial if there is no global ﬁxed point. If T is a simplicial real Fn-tree with trivial edge stabilizers, then the set of conjugacy classes of nontrivial vertex stabilizers of T is a free factor system denoted F(T ). The reformulation is as follows.

Theorem 5.1. For every ﬁnitely generated unipotent subgroup H of Out(Fn) there is a nontrivial simplicial Fn-tree T with all edge stabilizers trivial that is ﬁxed by all elements of H. Furthermore, there is such a tree with exactly one orbit of edges and if F is any maximal proper H-invariant free factor system then T may be chosen so that F(T )=F.

Such a tree can be obtained from the marked filtered graph produced by Theorem 1.1 by taking the universal cover and then collapsing all edges except for the lifts of the highest edge EK . For a proof of the reverse implication, namely that Theorem 5.1 implies Theorem 1.1, see Section 6. Along the way we obtain a result that is of interest in its own right. The necessary background material on trees may be found in Section 2, but also we give a quick review here. Simplicial Fn-trees may be endowed with metrics by equivariantly assigning lengths to edges. Given a simplicial real Fn-tree T and an element a ∈ Fn, the number T (a) is defined to be the infimum 6 MLADEN BESTVINA, MARK FEIGHN, AND MICHAEL HANDEL of the distances that a translates elements of T . It is through these length functions that the space of simplicial real Fn-trees is topologized. Again there is a natural right action of Out(Fn). We will work in the Out(Fn)-subspace T consisting of those nontrivial simplicial real trees that are limits of free actions.

Theorem 1.4 Suppose T ∈T and O∈UPG(Fn). There is an integer d = d(O,T) ≥ 0 such that the sequence {T Ok/kd} converges to a tree T O∞ ∈T.

This is proved in Section 4 as Theorem 4.22, which also contains an explicit description of the limit tree in the case that d(O,T) ≥ 1. Section 5 is the heart of the proof of Theorem 5.1. For notational sim- plicity, let us assume that H is generated by two elements, O1 and O2. Given T ∈T, let Elliptic(T ) be the subset of Fn consisting of elements fixing a point of T . Elements of Elliptic(T ) are elliptic. Choose T0 ∈T such that T0 has trivial edge stabilizers and such that Elliptic(T0)isH-invariant and maximal, i.e. such that if T ∈T has trivial edge stabilizers, if Elliptic(T )isH-invariant, and if Elliptic(T0) ⊂ Elliptic(T ), then Elliptic(T0) = Elliptic(T ). We prove that T0 satisfies the conclusions of Theorem 5.1 but not by a direct analysis of T0. Rather, we consider the “bouncing sequence” {T0,T1,T2, ···} T O∞ in defined inductively by Ti+1 = Ti i+1 where the subscripts of the outer automorphisms are taken mod 2. We establish properties of Ti for large i and then use these to prove that T0 is the desired tree. The key arguments in Section 5 are Proposition 5.5, Proposition 5.7, and Proposition 5.13. They focus not on discovering “ping-pong” dynamics (H may well contain a nonabelian free group), but rather on constructing an element in H of exponential growth. The connection to the bouncing sequence is as O∞O∞ O∞ follows. Properties of the tree Tk = T0 1 2 ... k−1 are reflected in the dy- O ON1 ON2 ONk−1 namics of the ‘approximating’ outer automorphism (k)= 1 2 ... k−1 where N1 N2 ··· Nk−1 1. We verify properties of Tk by proving that if the property did not hold, then O(k) would have exponential growth. After the breakthrough of E. Rips and the subsequent successful applica- tions of the theory by Z. Sela and others, it became clear that trees were the right tool for proving Theorem 1.1. Surprisingly, under the assumption that H is finitely generated (which is the case that we are concerned with in this paper and which suffices for proving that the Tits alternative holds), we only work with simplicial real trees and the full scale R-tree theory is never used. However, its existence gave us a firm belief that the project would succeed, and, indeed, the first proof we found of the Tits alternative used this theory. In a sense, our proof can be viewed as a development of the program, started by Culler-Vogtmann [CV86], to use spaces of trees to understand Out(Fn)in much the same way that Teichmüller space and its compactification were used by Thurston and others to understand mapping class groups. THE TITS ALTERNATIVE FOR Out(Fn) II 7

2. Fn-trees

In this section, we collect the facts about real Fn-trees that we will need. This paper will only use these facts for simplicial real trees, but we sometimes record more general results for anticipated later use. Much of the material in this section can be found in [Ser80], [SW79], [CM87], or [AB87].

2.1. Real trees. An arc in a topological space is a subspace homeomorphic to a compact interval in R. A point is a degenerate arc. A real tree is a metric space with the property that any two points may be joined by a unique arc, and further, this arc is isometric to an interval in R (see for example [AB87] or [CM87]). The arc joining points x and y in a real tree is denoted by [x, y]. A branch point of a real tree T is a point x ∈ T whose complement has other than 2 components. A real tree is simplicial if it is equipped with a discrete subspace (the set of vertices) containing all branch points such that the edges (closures of the components of the complement of the set of vertices) are compact. If the subspace of branch points of a real tree T is discrete, then it admits a (nonunique) structure as a simplicial real tree. The simplicial real trees appearing in this paper will come with natural maps to compact graphs and the vertex sets of the trees will be the preimages of the vertex sets of the graphs. For a real tree T , a map σ : J → T with domain an interval J is a path in T if it is an embedding or if J is compact and the image is a single point; in the latter case we say that σ is a trivial path. If the domain J of a path σ is compact, define the inverse of σ, denoted σ or σ−1,tobeσ ◦ ρ where ρ : J → J is a reflection. We will not distinguish paths in T that differ only by an orientation- preserving change of parametrization. Hence, every map σ : J → T with J compact is properly homotopic rel endpoints to a unique path [σ] called its tightening. If σ : J → T is a map from the compact interval J to the simplicial real tree T and the endpoints of J are mapped to vertices, then the image of [σ], if nondegenerate, has a natural decomposition as a concatenation E1 ···Ek where each Ei,1≤ i ≤ k, is a directed edge of T . The sequence E1 ···Ek is called the edge path associated to σ. We will identify [σ] with its associated edge path. This notation extends naturally if the domain of the path is a ray or the entire line and σ is an embedding. A path crosses an edge of T if the edge appears in the associated edge path. A path is contained in a subtree if it crosses only edges of the subtree. A ray in T is a path [0, ∞) → T that is an embedding.

2.2. Real Fn-trees. By Fn denote a fixed copy of the free group with basis {e1,...,en}. A real Fn-tree is a real tree equipped with an action of Fn by 8 MLADEN BESTVINA, MARK FEIGHN, AND MICHAEL HANDEL isometries. It is minimal if it has no proper Fn-invariant subtrees. If H is a subgroup of Fn then FixT (H) denotes the subset of T consisting of points that are fixed by each element of H.Ifa ∈ Fn, then FixT (a):=FixT (a). If X ⊂ T , then StabT (X) is the subgroup of Fn consisting of elements that leave X invariant. If x ∈ T , then StabT (x) := StabT ({x}). The symbol ‘[[·]]’ denotes ‘conjugacy class’. Define

Point(T ):={[[StabT (x)]] | x ∈ T, StabT (x) = 1} and

Arc(T ):={[[StabT (σ)]]

| σ is a nondegenerate arc in T, StabT (σ) = 1}.

The length function of a real Fn-tree T assigns to a ∈ Fn the number

T (a):=infx∈T {dT (x, ax)}.

Length is constant on conjugacy classes, so we also write T ([[a]]) for T (a). If T (a) is positive, then a (or [[a]]) is hyperbolic in T , otherwise a is elliptic.If a is hyperbolic in T , then {x ∈ T | dT (x, ax)=T (a)} is isometric to R. This set is called the axis of a and is denoted AxisT (a). The restriction of a to its axis is translation by T (a). If a is elliptic in T then a ﬁxes a point of T . Thus, an element of Fn is in Elliptic(T ) if it is trivial or if its conjugacy class is in Point(T ). A subgroup of Fn is elliptic if all elements are elliptic. A real Fn-tree T is trivial if FixT (Fn) = ∅. In particular, a minimal tree is trivial if and only if it is a point. We will need the following special case of a result of Serre.

Theorem 2.1 ([Ser80]). Suppose that T is a real Fn-tree where Fn = a1,...,ak. Suppose that aiaj is elliptic in T for 1 ≤ i, j ≤ k. Then T is trivial. 2.3. Very small trees. We will only need to consider a restricted class of real trees. A real Fn-tree T is very small [CL95] if (1) T is nontrivial,

(2) T is minimal.

(3) The subgroup of Fn of elements pointwise ﬁxing a nondegenerate arc of T is either trivial or maximal cyclic, and

(4) for each 1 = a ∈ Fn, FixT (a) is either empty or an arc. It follows from (3) that if T is very small and x, y ∈ T , then each element of StabT ([x, y]) ﬁxes [x, y] pointwise. In particular, if T is simplicial, then no element of Fn inverts an edge. THE TITS ALTERNATIVE FOR Out(Fn) II 9

We will need:

Theorem 2.2 ([CM87], [AB87]). Let Q be a ﬁnitely generated group, and let T be a minimal nontrivial Q-tree. Then the axes of hyperbolic elements of Q cover T .

In the case of simplicial trees, the following theorem is established by an easy Euler characteristic argument. The generalization to R-trees due to Gaboriau and Levitt uses more sophisticated techniques.

Theorem 2.3 ([GL95]). Let T be a very small Fn-tree. There is a bound depending only on n to the number of conjugacy classes of point and arc stabilizers. The rank of a point stabilizer is no more than n with equality if and only if T/Fn is a wedge of circles and each edge of T has inﬁnite cyclic stabilizer.

2.4. Spaces of real Fn-trees. Let R+ denote the ray [0, ∞) and let C denote the set of conjugacy classes of elements in Fn. The space Tall of nontrivial minimal real Fn-trees is given the smallest topology such that the map T → RC θ : all +, given by θ(T )=(T (a))[[a]]∈C is continuous. Let TCV denote the subspace of Tall consisting of free simplicial actions. The closure of TCV in Tall is denoted TVS. The subspace of simplicial trees in TVS is denoted T . The map θ is injective when restricted to TVS; see [CM87]. In other words, if S, T ∈TVS satisfy θ(S)=θ(T ), then S and T are equivariantly isometric. In this paper, we only need to work in T although some results are presented in greater generality. The automorphism group Aut(Fn) acts naturally on Tall on the right by twisting the action; i.e., if the action on T ∈Tall is given by (a, t) → a · t and if Φ ∈ Aut(Fn) then the action on T Φ is given by (a, t) → Φ(a)·t. In terms of the length functions, the action is given by T Φ(a)=T (Φ(a)) for Φ ∈ Aut(Fn), T ∈Tall, and a ∈ Fn. The subgroup Inner(Fn) of inner automorphisms acts trivially, and we have an action of Out(Fn) = Aut(Fn)/Inner(Fn). The spaces TCV , TVS, and T are all Out(Fn)-invariant. To summarize, for O∈Out(Fn) and T ∈TVS, the following are equivalent.

•Oﬁxes T .

• T (O([[γ]])) = T ([[γ]]) for all γ ∈ Fn.

• For any Φ ∈ Aut(Fn) representing O, there is a Φ-equivariant isometry fΦ : T → T .

2.5. Bounded cancellation constants. We will often need to compare the length of the same element of Fn in diﬀerent real Fn-trees. This is facilitated by the existence of bounded cancellation constants. 10 MLADEN BESTVINA, MARK FEIGHN, AND MICHAEL HANDEL

Deﬁnition 2.4. Let S and T be real Fn-trees. The bounded cancellation constant of an Fn-map f : S → T , denoted BCC(f), is the least upper bound of numbers B with the property that there exist points x, y, z ∈ S with y ∈ [x, z] so that the distance between f(y) and [f(x),f(z)] is B.

Cooper [Coo87] showed that if S and T are in TCV and if f is PL, then BCC(f) is ﬁnite. For a map f : X → Y between metric spaces we denote by Lip(f) the Lipschitz constant of f; i.e.,

Lip(f):=sup{dY (f(x1),f(x2))/dX (x1,x2) | (x1,x2) ∈ X × X, x1 = x2}. The map f is Lipschitz if Lip(f) < ∞. The following generalization of Cooper’s result is an immediate consequence of Lemma 3.1 of [BFH97a].

Proposition 2.5. Suppose that S ∈TCV , T ∈TVS, and f : S → T is a Lipschitz Fn-map. Then, BCC(f) < ∞.

2.6. Real graphs. In [BFH00], marked graphs were used. Here we will need graphs with a metric structure. A real graph is a locally finite graph (one-dimensional CW-complex) whose universal cover has the structure of a simplicial real tree with covering transformations acting by isometries. A locally finite graph with specified edge lengths determines a real graph. Occasionally, it is convenient to view a locally finite graph as a real graph. To do this, we will specify edge lengths. If no lengths are mentioned, then they are assumed to be 1. Let G be a real graph with universal covering p :Γ→ G. A map σ : J → G with domain an interval J is a path if σ = p ◦ σ˜ whereσ ˜ is a path in Γ. The terminology for paths in trees transfers directly over to real graphs; cf. [BFH00, p. 525]. A closed path in G is a path whose initial and terminal endpoints coincide. A circuit is an immersion from the circle S1 to G; homotopic circuits are not distinguished. Any homotopically nontrivial map σ : S1 → G is homotopic to a unique circuit [[σ]]. Circuits are identified with cyclically ordered edge paths which we call associated edge circuits. A circuit crosses an edge if the edge appears in the circuit’s associated edge circuit. A circuit is contained in a subgraph if it crosses only edges of the subgraph. We make standard identifications between based closed paths and elements of the fundamental group and between circuits and conjugacy classes in the fundamental group. A marked real graph is a real graph G together with a homotopy equivalence µ : Rosen → G. The universal cover of a marked real graph has a structure of a real free Fn-tree that is well-defined up to equivariant isometry. A real Fn-tree T admits an Fn-equivariant mapµ ˜ : Rosen → T . This map is well-defined up to equivariant homotopy. If the action is free, then the quotient µ : Rosen → G is a marking. THE TITS ALTERNATIVE FOR Out(Fn) II 11

A core graph is a ﬁnite graph with no vertices of valence 1 or 0. Any connected graph with ﬁnitely generated fundamental group has a unique maximal core subgraph, called its core. The core of a forest is empty.

2.7. Models and normal forms for simplicial Fn-trees. References for this section are [SW79] and [Ser80]. A map h : Y → Z with Y and Z CW-complexes is cellular if, for all k, the k-skeleton of Y maps into the k- (k) (k) skeleton of Z, i.e. h(Y ) ⊂ Z . Given CW-complexes Y , Z0, and Z1 and cellular maps gi : Y → Zi, the double mapping cylinder D(g, h)ofg and h is the quotient (Y × [0, 1]) (Z0 Z1)/ ∼ where ∼ is the equivalence relation generated by (y, 0) ∼ g0(y) and (y, 1) ∼ g1(y). The double mapping cylinder is naturally a CW-complex with a map to [0, 1]. In the case where Z0 = Z1, we modify the definition of D(g, h) so that corresponding points of Z0 and Z1 are also identified. In this case, D(g, h) has a natural map to S1. Let Rosen denote a fixed wedge of n oriented circles with a fixed identifi- th cation of π1(Rosen, ∗) with Fn such that the i circle corresponds to ei. Also fix a compatible identification of Fn with the covering transformations of the universal cover Rosen of Rosen. Let T be a simplicial real Fn-tree and let T denote the real graph T/Fn. A graph of spaces over T is a CW-complex X with a cellular map q : X → T such that: • For each vertex x of T , q−1(x) is a subcomplex of X.

• For each edge e of T with endpoints v and w (possibly equal), there is a −1 CW-complex Xe, a pair of cellular maps g : Xe → q (v) and h : Xe → q−1(w), and isomorphisms D(g, h) → q−1(e) and S1 or [0, 1] → e such that the following diagram commutes. D(g, h) q−1(e)

S1or [0, 1] e

A vertical subspace of X is a subcomplex of the form q−1(x) for some vertex x ∈ T . An edge of a vertical subspace is vertical. Other edges of X are horizontal. An edge path consisting of vertical edges is vertical.

2.6. × × Example The quotient Rosen Fn T of Rosen T by the diagonal × → action of Fn with the map Q : Rosen Fn T T induced by projection onto the second coordinate is naturally a graph of spaces over T .Itisan Eilenberg-MacLane space. Its fundamental group is naturally identiﬁed with Fn by the map to Rosen induced by projection onto the ﬁrst coordinate. If x −1 × is a vertex of T , then Q (x) is a full subcomplex of Rosen Fn T isomorphic 12 MLADEN BESTVINA, MARK FEIGHN, AND MICHAEL HANDEL

−1 to Rosen/StabT (˜x) wherex ˜ is a lift of x to T . In particular, Q (x) is a graph homotopy equivalent to the wedge Rosex of nx circles where nx is the rank of StabT (˜x). Similarly, if x is a point in the interior of an edge e of T , then the preimage of x is isomorphic to Rosen/StabT (˜e) wheree ˜ is a lift of e to T .In −1 particular, Q (x) is a graph homotopy equivalent to the wedge Rosee of ne circles where ne is the rank of StabT (˜e).

Let M be the set of midpoints of edges of T .Amodel for T is a graph of × → spaces X over T with a homotopy equivalence Rosen Fn T X such that the following diagram commutes up to a homotopy supported over the complement of M: × Rosen Fn T Q q X T and such that the induced map Q−1(M) → q−1(M) is a homotopy equivalence. ← × → The homotopy equivalence X Rosen Fn T Rosen identiﬁes conjugacy classes in π1(X) with conjugacy classes in Fn and is called the induced marking. The trees in this paper will all be minimal with ﬁnitely generated vertex and edge stabilizers. (In fact, edge stabilizers will be cyclic.) Until Section 5.4, we will make the following additional requirements of our models. • If x is a vertex of T , then the vertical subspace q−1(x) is a subcomplex of X isomorphic to Rosex.

• If e is an edge of T , then Xe is isomorphic to Rosee. Models satisfying these properties are constructed in [SW79].

Example 2.7. Pictured below is an example of a model X together with the induced marking µ : Rose2 → X and the quotient map q : X → T . The µ-image of the edge ‘e1’ is the edge ‘a’ and the µ-image of the edge ‘e2’isthe only horizontal edge ‘t’. The vertical space is a wedge of two circles. This model satisﬁes all of the above properties.

e2 t µ q

Rose2 X(T ) T − e1 a [t 1at] THE TITS ALTERNATIVE FOR Out(Fn) II 13

Any path in X whose endpoints are vertices is homotopic rel endpoints to an edge path in X(1) of the form

ν0H1ν1H2ν2 ···Hmνm where νi is a (possibly trivial) vertical edge path and Hi is a horizontal edge of X. The length of the path is the sum of the lengths q(Hi). Such an edge path is in normal form unless for some i we have that HiνiHi+1 is homotopic rel endpoints into a vertical subspace. If σ is a path in X whose endpoints are vertices, then [σ] is an edge path homotopic rel endpoints to σ that is in normal form. If the displayed edge path is not in normal form and if i is as above, then the path is homotopic to the path obtained by replacing νi−1HiνiHi+1νi+1 by a path in a vertical subspace that is homotopic rel endpoints. We call this process erasing a pair of horizontal edges. Any edge path in X may be put into normal form by iteratively erasing pairs horizontal edges (see [SW79]). Two paths in normal form that are homotopic rel endpoints have the same length. In an analogous fashion, circuits in X have lengths and normal forms. If σ is a circuit in X, then [[σ]] is a circuit freely homotopic to σ that is in normal form. Note that lengthX ([[σ]]) = T ([[a]]) where [[a]] is the conjugacy class of Fn represented by the image of σ under the induced marking of X. If σ1 and σ2 are paths in X with the same initial points, then the overlap length of σ1 and σ2 is deﬁned to be 1 · length ([σ ]) + length ([σ ]) − length ([σ σ ]) . 2 X 1 X 2 X 1 2

Remark 2.8. Suppose that σ1,σ2 and σ3 are paths in X with endpoints at vertices, that the terminal endpoint of σ1 is the initial endpoint of σ2 and that the terminal endpoint of σ2 is the initial endpoint of σ3. Let D be the overlap length ofσ ¯1 and σ2, let D be the overlap length ofσ ¯2 and σ3 and assume that lengthX ([σ2]) >D+ D . In the proof of Proposition 4.21 we use the fact, immediate from the deﬁnitions, that the following quantities are realized as lengths of edge paths in T .

• lengthX ([σ2]). • − lengthX ([σ2]) (D + D ).

Example 2.9. Let X be as in Example 2.7. Then the overlap length of t and at is the same as the length in X of t even though the maximal common initial segment of t and at is degenerate. Of course, at = t[t−1at] are both normal forms and the maximal common initial segment of t and t[t−1at]ist. 14 MLADEN BESTVINA, MARK FEIGHN, AND MICHAEL HANDEL

2.8. Free factor systems. Here we review deﬁnitions and background for free factor systems as treated in [BFH00]. i 1 2 k We reserve the notation F for free factors of Fn.IfF ∗ F ∗···∗F is a free factor and each F i is nontrivial (and so has positive rank), then we say that the collection F = {[[F 1]], [[F 2]],...,[[F k]]} is a nontrivial free factor system. We refer to ∅ as the trivial free factor system. A free factor system F is proper if it is not {[[Fn]]}. We write [[F 1]] [[F 2]] if F 1 is conjugate to a free factor of F 2 and write i j i j F1 F2 if for each [[F ]] ∈F1 there exists [[F ]] ∈F2 such that [[F ]] [[F ]]. We say that F1 F2 is proper if F1 = F2. The next lemma follows immediately from Lemma 2.6.3 of [BFH00].

Lemma 2.10. There is a bound, depending only on n, to the length of a chain F1 F2 ··· FN of proper ’s.

We say that a subset X of Fn is carried by the free factor system F if X ⊂ F i for some [[F i]] ∈F. A collection X of subsets is carried by F if each X ∈X is carried by some element of F. Let ∂Fn denote the boundary of Fn. Let Rn (for rays) denote the quotient of ∂Fn by the action of Fn. The natural action of Aut(Fn)on∂Fn descends to an action of Out(Fn)onRn.IfG is a marked real graph, then Rn is naturally identiﬁed with the set of rays in G where two rays are equivalent if their associated edge paths have a common tail. In [BFH00], a parallel treatment was given using lines instead of rays. The reader is referred there for details. i i A free factor F of Fn gives rise to a subset R of Rn. In terms of a tree i T ∈TCV , a ray represents an element of R if it can be Fn-translated so that i its image is eventually in the minimal F -subtree of T . A ray R ∈Rn is carried by F i if R ∈Ri.Itiscarried by the free factor system F if it is carried by i i F for some [[F ]] ∈F. A subset of Rn is carried by F if each element of the subset is carried by some element of F. The proof of the following lemma is completely analogous to the proof of Corollary 2.6.5 of [BFH00].

Lemma 2.11. Let X be a collection of subsets of Fn and let R be a subset of Rn. Then there is a unique minimal (with respect to ) free factor system F that carries both X and R.

A (not necessarily connected) subgraph K of a marked real graph determines a free factor system F(K) as in [BFH00, Ex. 2.6.2]. If T is a simplicial real Fn-tree with trivial edge stabilizers, then the set of conjugacy classes of nontrivial vertex stabilizers of T is a free factor system denoted F(T ). THE TITS ALTERNATIVE FOR Out(Fn) II 15

3. Unipotent polynomially growing outer automorphisms

In this section we bring outer automorphisms into the picture. We will consider a class of outer automorphisms that is analogous to the class of unipotent matrices. First we review the linear algebra of unipotent matrices.

3.1. Unipotent linear maps. The results in this section are standard. We include proofs for the reader’s convenience. Throughout this section, R denotes either Z or C, and V denotes a free R-module of ﬁnite rank.

Proposition 3.1. Let f : V → V be an R-module endomorphism. The following conditions are equivalent: (1) V has a basis with respect to which f is upper triangular with 1’s on the diagonal. (2) (Id − f)rank(V ) =0. (3) (Id − f)n =0for some n>0.

Proof. It is clear that (1) implies (2) and that (2) implies (3). To see that (3) implies (1), assume that (Id − f)n = 0. We may assume that W := Im(Id − f)n−1 = 0. The restriction of Id − f to the submodule W is 0, and hence each 0 = v ∈ W is fixed by f. After perhaps replacing v byarootin the case R = Z, we may assume that v is an f-fixed basis element of V . The proof now concludes by induction on rank(V ) using the fact that the induced homomorphism f : V/v→V/v also satisfies (Id − f )n =0.

An endomorphism f satisfying any of the equivalent conditions of Propo- sition 3.1 is said to be unipotent.

Corollary 3.2. Let f : V → V be an R-module endomorphism, and let W be an f-invariant submodule of V which is a direct summand of V . Then f is unipotent if and only if both the restriction of f to W and the induced endomorphism on V/W are unipotent.

Proof. The proof is evident if we use Proposition 3.1(1) in the “if” direction and Proposition 3.1(2) in the “only if” direction.

Corollary 3.3. Let f : V → V be unipotent. If x ∈ V is f-periodic, i.e. if f m(x)=x for some m>0, then x is f-ﬁxed, i.e. f(x)=x.

Proof. First assume that R = C. We may assume that − V = span(x, f(x), ··· ,fm 1(x)). m Let e1,e2,...,em be the standard basis for C . There is a surjective linear m i−1 map π : C → V given by π(ei)=f (x), and f lifts to the linear map 16 MLADEN BESTVINA, MARK FEIGHN, AND MICHAEL HANDEL

m m f : C → C , f(ei)=ei+1 mod m.Forλ ∈ C, the generalized λ-eigenspace is deﬁned to be {x ∈ Cm|(λI − f)m(x)=0}.

Since f is unipotent, the linear map π must map the generalized 1-eigenspace onto V (and all other generalized eigenspaces to 0). The characteristic polynomial λm − 1off has m distinct roots. In particular, the generalized 1-eigenspace is one-dimensional (and equals the 1-eigenspace of f). It follows that dim(V ) ≤ 1 and f(x)=x. If R = Z, just tensor with C.

Corollary 3.4. Let f : V → V be unipotent. If W is a direct summand which is periodic (i.e. f m(W )=W for some m>0), then W is invariant (i.e. f(W )=W ).

Proof. The restriction of f m to W is unipotent, so there is a basis element x ∈ W ﬁxed by f m. By Corollary 3.3, f(x)=x. The proof concludes by induction on rank(W ).

Proposition 3.5. Let A ∈ GLn(Z) have all eigenvalues on the unit circle (i.e. A grows polynomially). If the image of A in GLn(Z/3Z) is trivial, then A is unipotent.

Proof. We ﬁrst argue that some power AN of A is unipotent, i.e. that all eigenvalues of A are roots of unity. Choose N so that all eigenvalues of AN are close to 1. Then tr(AN ) is an integer close to n, and thus all eigenvalues of AN are equal to 1. n1 ··· nm Let f = f1 fm be the minimal polynomial for A factored into irre- Z ni ducibles in [x]. Let Ai = fi (A) and Ki = Ker(Ai). First note that each Ki = 0. For example, Im(A2A3 ···Am) ⊂ K1 but A2A3 ···Am = 0 since f is minimal. If A is not unipotent, then some fi,sayf1, is not x − 1. Since all roots of f are roots of unity, f1 is the minimal polynomial for a nontrivial root of unity and so it divides 1 + x + x2 + ···+ xr−1 for some r>1. The matrix 2 ··· r−1 st I + A + A + + A has nontrivial kernel (since its n1 power vanishes r on K1). A nonzero integral vector v in this kernel satisﬁes A (v)=v and A(v) = v. Then Fix(Ar) is a nontrivial direct summand of Zn, the restriction of A to this summand is nontrivial and periodic, and the induced endomorphism of Fix(Ar) ⊗ Z/3Z is the identity. This contradicts the standard fact that the kernel of GLk(Z) → GLk(Z/3Z) is torsion-free.

3.2. Topological representatives. A homotopy equivalence f : G → G of a marked real graph induces an outer automorphism O of Fn via the fixed identification of Fn with the fundamental group of Rosen.Iff maps vertices THE TITS ALTERNATIVE FOR Out(Fn) II 17 to vertices and if the restriction of f to each edge of G is an immersion, then we say that f is a topological representative of O. A filtration for a topological representative f : G → G is an increasing sequence of f-invariant subgraphs ∅ = G0 G1 ··· GK = G. The closure th of Gr \ Gr−1 is called the r stratum. If the path σ = σ1σ2 is the concatenation of paths σ1 and σ2, then σ i i i splits, denoted σ = σ1 · σ2,if[f (σ)] = [f (σ1)][f (σ2)] for all integers i ≥ 0; see [BFH00, pp. 553–554]. In this paper, as in [BFH00], it is critically important to understand the behavior of paths under iteration by f. If a path splits, the behavior of the path is determined by the behavior of the subpaths.

3.3. Relative train tracks and automorphisms of polynomial growth. The techniques of this paper depend on being able to ﬁnd good representatives for outer automorphisms of polynomial growth.

Deﬁnition 3.6. An outer automorphism O∈Out(Fn) has polynomial growth if, given a ∈ Fn, there is a polynomial P ∈ R[x] such that the (reduced) word length of Oi([[a]]) is bounded above by P (i). The set of outer automorphisms having polynomial growth is denoted PG(Fn) (or just PG).

It follows from [BH92] that the deﬁnition of polynomial growth given above agrees with the deﬁnition on page 564 of [BFH00]. We start by recalling the topological representatives for automorphisms having polynomial growth that were found in [BH92].

Theorem 3.7 ([BH92]). An automorphism O∈PG(Fn) has a topological representative f : G → G with a ﬁltration ∅ = G0 G1 ··· GK = G such that

(1) for every edge E ∈ Gi \ Gi−1, the edge path f(E) crosses exactly one edge in Gi \ Gi−1 and it crosses that edge exactly once.

(2) If F is an O-invariant free factor system, it can be arranged that F = F(Gr) for some r. If O is the identity on each conjugacy class in F, it can be arranged that f = Id on Gr.

Deﬁnition 3.8. A topological representative as in Theorem 3.7 is called a relative train track (RTT) representative for O.

3.4 Unipotent representatives and UPG automorphisms.

Deﬁnition 3.9. An outer automorphism is unipotent if it has polynomial growth and its action on H1(Fn; Z) is unipotent. The set of unipotent automorphisms is denoted by UPG(Fn) (or just UPG). 18 MLADEN BESTVINA, MARK FEIGHN, AND MICHAEL HANDEL

We now recall a special case of an improvement of RTT representatives from [BFH00].

Deﬁnition 3.10. Let f : G → G be an RTT representative. A nontrivial path τ in G is a periodic Nielsen path if, for some m ≥ 0, [f m(τ)]=[τ]. If m = 1 then τ is a Nielsen path.Anexceptional path in G is a path of the form m Eiτ Ej where Gi \ Gi−1 is the single edge Ei, Gj \ Gj−1 is the single edge Ej, p q τ is a Nielsen path, f(Ei)=Eiτ , and f(Ej)=Ejτ for some m ∈ Z, p, q > 0.

Theorem 3.11 ([BFH00, Th. 5.1.8]). Suppose that O∈UPG(Fn) and that F is an O-invariant free factor system. Then there is an RTT representative f : G → G and a ﬁltration ∅ = G0 G1 ··· GK = G representing O with the following properties:

(1) F = F(Gr) for some ﬁltration element Gr.

(2) Each Gi \ Gi−1 is a single edge Ei satisfying f(Ei)=Ei · ui for some closed path ui with edges in Gi−1.

(3) Every vertex of G is ﬁxed by f.

(4) Every periodic Nielsen path has period one.

(5) If σ is any path with endpoints at vertices, then there exists M = M(σ) so that for each m ≥ M,[f m(σ)] splits into subpaths that are either single edges or are exceptional.

(6) M(σ) is a bounded multiple of the edge length of σ.

Remark 3.12. Another useful condition is

(7) If Ei and Ej are distinct edges of G with nontrivial suﬃxes ui and uj, then ui = uj.

This property is part of the construction of f : G → G from [BFH00, Th. 5.1.8]. There is an operation called sliding that is used for nonexponentially growing strata. Condition 1 of [BFH00, Prop. 5.4.3] implies Item (7). Alternatively, starting with f : G → G satisfying (1–6), f may be enhanced to also satisfy (7) by replacing Ej with EjE¯i.

Deﬁnition 3.13. An RTT representative f satisfying Items (1–7) above is a unipotent representative or a UR. The based closed paths ui are suﬃxes of f.

Remark 3.14. Note that Item (2) can be restated as

k k−1 [f (Ei)] = Ei · ui · [f(ui)] ·····[f (ui)] THE TITS ALTERNATIVE FOR Out(Fn) II 19 for all k>0. Since exceptional paths do not have nontrivial splittings, the k splitting of [f (Ei)] guaranteed by Item (5) restricts to a splitting of ui into single edges and exceptional paths. The immersed inﬁnite ray k−1 Ri = Eiui[f(ui)] ···[f (ui)] ··· is the eigenray associated to Ei. Lifts of Ri to the universal cover of G are m also called eigenrays. The subpaths [f (ui)] of Ri are sometimes referred to as blocks.

For example, the map f : G → G on the wedge of two circles with edges a and b given by f(a)=a, f(b)=ba is a UR. For ω = ba−10bab−1 we may take M(ω) = 10 in Item (5), since [f 10(ω)] = b · (bab−1) is a splitting into an edge and an exceptional (Nielsen) path. The map given by a → a, b → ba, c → cba−1 on the wedge of three circles is not a UR since ω = cba−1 does not eventually split as in Item (5). Replacing ba−1 by b yields a UR of the same outer automorphism.

Deﬁnition 3.15. Let f : G → G be a UR with ﬁltration ∅ = G0 G1 ··· GK = G. The highest edge (or stratum) of G is EK = GK \ GK−1. The height of a path σ in G, denoted height(σ), is the smallest m such that the path crosses only edges in Gm.Ifσ is a path of height m, then a highest edge in σ is an occurrence of Em or Em in σ. By [BFH00, Lemma 4.1.4], the path σ naturally splits at the initial endpoints of its highest edges; this is called the highest edge splitting of σ.

Many arguments in this paper are inductions on height.

Proposition 3.16. If O∈UPG(Fn), then all O-periodic conjugacy classes are ﬁxed.

Proof. Let f : G → G beaURforO and let σ be a circuit in G representing an O-periodic conjugacy class. Consider the highest edge splitting of σ. Each of the resulting subpaths is an f-periodic Nielsen path. Theorem 3.11(4) now implies that each subpath is f-ﬁxed, and thus σ is f-ﬁxed.

We will also need the following more technical results.

Lemma 3.17 ([BFH00, Lemma 5.7.9]). Suppose that f : G → G is a UR. There is a constant C so that if ω is a closed path that is not a Nielsen path, σ = αωkβ is a path, and k>0, then at most C copies of [f m(ω)] are canceled when [f m(α)][f m(ωk)][f m(β)] is tightened to [f m(σ)].

The following proposition is the analogue of the fact in linear algebra that if A is a unipotent matrix and v a nonzero vector, then projectively the sequence {Ak(v)} converges to an eigenspace of A. 20 MLADEN BESTVINA, MARK FEIGHN, AND MICHAEL HANDEL

Proposition 3.18. Let f : G → G be a UR with edge Ei and suﬃx ui. If ∗ [f(ui)] = ui, R is an initial segment of Ri, and σ is a path in G that crosses k Ei or its inverse, then there is an N such that, for all k>N,[f (σ)] contains R∗ or its inverse as a subpath.

Proof. We argue by induction on height(σ). If height(σ)=i, consider the splitting of [f M (σ)] into edges and exceptional paths (see Theorem 3.11(5)). M There is a 1-1 correspondence between occurrences of Ei in σ and in [f (σ)]. Since [f(ui)] = ui, Ei does not occur in an exceptional path, and hence one of ∗ the subpaths in the splitting is Ei or Ei. Eventually, the iterates contain R or its inverse. Now assume height(σ)=j>i. Again consider the splitting of [f M (σ)] into edges and exceptional paths. We first claim that an exceptional path k Esτ Et cannot cross Ei or Ei. Indeed, suppose that this exceptional path does cross Ei or Ei. It must be then that τ crosses Ei or its inverse because the edges Es and Et have fixed suffixes and so are distinct from Ei and Ei. But, τ cannot cross Ei of Ei for otherwise, since height(τ)

4. The dynamics of unipotent automorphisms

4.1. Poloynomial sequences. Suppose T ∈T and O∈UPG(Fn). Our goal in this section is to show that there is a natural number d = d(O,T) such that { Ok d}∞ O∞ ∈T the sequence T /k k=0 converges to a tree T . This is the content of Theorem 4.22. Theorem 4.22 will be proved by showing that if f : G → G is a UR for O, if h : G → X is a homotopy equivalence from G to a model for T taking vertices to vertices, and if σ is a path in G with endpoints at vertices, then THE TITS ALTERNATIVE FOR Out(Fn) II 21 there is a polynomial P such that, for large k, the length of [h(f k(σ))] equals P (k). Theorem 3.11 completely describes the [f k(σ)]’s. To measure the length of [h(f k(σ))], we must ﬁrst transfer f k(σ)toX via h and then put this path into normal form. The main work is in understanding the cancellation that occurs when [h(f k(σ))] is put into normal form. All paths will be assumed to have endpoints that are vertices. k The key properties of a sequence of paths {[f (σ)]}k are captured in the following deﬁnition.

Deﬁnition 4.1. Let G be a real graph. A sequence of paths in G is polynomial if it can be obtained from constant sequences of paths by ﬁnitely many operations of the following four basic types. { }∞ (1) (re-indexing and truncation): The sequence of paths Ak k=k is ob- { }∞ 0 tained from the sequence of paths Bk k=k by re-indexing and trunca- 1 tion if there is an integer k ≥ k1 − k0 such that Ak = Bk+k . (2) (inversion): The sequence of paths {A }∞ is obtained from the se- k k=k0 quence of paths {B }∞ by inversion if A is the inverse of B . k k=k0 k k (3) (concatenation): The sequence of paths {A }∞ is obtained from the k k=k0 sequences of paths {B }∞ and {C }∞ by concatenation if A = k k=k0 k k=k0 k BkCk. (As the notation implies, no cancellation occurs in BkCk.) (4) (integration): The sequence of paths {A }∞ is obtained from the se- k k=k0 quence of paths {B }∞ by integration if k k=k0 ··· Ak = Bk0 Bk0+1 Bk. (Again no cancellation occurs.)

For example, in a wedge of three circles with edges A, B, and C, the sequences {ABkC} and {ABAB2AB3 ···ABk} are polynomial. A sequence eventually has a property if it may be truncated and re-indexed so that the resulting sequence has the property. The elements of a sequence eventually have a property if only ﬁnitely many elements do not have the property.

Lemma 4.2. Let f : G → G be a UR of a unipotent automorphism. Let { k }∞ σ be a path in G. Then the sequence [f (σ)] k=0 is eventually polynomial. Proof. We use induction on the height of σ. If the height is 1, the sequence is constant. For the induction step, replace σ by the iterate [f M (σ)] from Theorem 3.11 so that it splits into subpaths which are either single edges or exceptional paths. It suﬃces to prove the statement for these subpaths. The statement is clear for the exceptional subpaths. For a single edge E with 22 MLADEN BESTVINA, MARK FEIGHN, AND MICHAEL HANDEL f(E)=E · u, the sequence {[f k(E)]} is the concatenation of the constant sequence {E} with the integral of the sequence {[f k(u)]} by Theorem 3.11(2).

The complexity of a polynomial sequence {Ak}, denoted

complexity({Ak}),

is the minimal number of basic operations needed to make {Ak}. The complexity of a constant sequence is 0. To measure the lengths of polynomial sequences of paths, we have polynomial sequences of numbers.

Deﬁnition 4.3. A sequence of nonnegative real numbers is polynomial if it can be obtained from constant sequences of nonnegative real numbers by ﬁnitely many operations of the following three basic types. { }∞ (1) (re-indexing and truncation): The sequence pk k=k is obtained from { }∞ 0 the sequence qk k=k by re-indexing and truncation if there is an integer 1 k ≥ k1 − k0 such that pk = qk+k . (2) (concatenation): The sequence {p }∞ is obtained from the sequences k k=k0 {q }∞ and {r }∞ by concatenation if p = q + r . k k=k0 k k=k0 k k k (3) (integration): The sequence {p }∞ is obtained from the sequence k k=k0 {q }∞ by integration if k k=k0 ··· pk = qk0 + qk0+1 + + qk.

The following lemma is immediate from the deﬁnitions.

Lemma 4.4. Let {A }∞ be a polynomial sequence of paths in G. Then k k=k0 the sequence of nonnegative real numbers {length (A )}∞ is polynomial. G k k=k0

The complexity of a polynomial sequence {pk}, denoted

complexity({pk}), is the minimal number of basic operations needed to make {pk}. The complexity of a constant sequence is 0.

Lemma 4.5. (1) If 0 is an element of a polynomial sequence of nonnegative real numbers, then the sequence is constantly 0.

(2) Unless a polynomial sequence of nonnegative real numbers is constant, it is increasing.

(3a) If {pk} is a polynomial sequence of real numbers, then there is a polynomial P ∈ R[x] such that P (k)=pk. THE TITS ALTERNATIVE FOR Out(Fn) II 23

(3b) If {pk} is not constant, if {mj,k}k are the positive constant sequences used in the integration operations in a particular construction of {pk}, if m = minj{mj,k}, and if P has degree d, then the leading coeﬃcient of P is bounded below by m/d!.

(4) If {pk} is a polynomial sequence of nonnegative real numbers and if c ∈ R is eventually not greater than pk, then the sequence {pk −c} is eventually polynomial.

Proof. In each case, the proof is by induction on complexity.

(1) The statement is true for constant sequences. If qk and rk are never 0, ··· then the same is true for qk + rk and qk0 + + qk. (2) The sum of constant sequences is constant. The sum of increasing and constant sequences is increasing as long as at least one of the sequences is increasing.

(3) The proof is an induction on the complexity of {pk}.If{pk} is constant then P (k)=pk for a constant polynomial P . Suppose {qk} and {rk} are polynomial sequences, that Q and R are polynomials with Q(k)=qk and R(k)=rk, that the leading coefficients of Q and R are respectively Q0 and R0, that deg(Q) ≥ deg(R), and that deg(Q) ≥ 1. If {pk} is obtained from {qk} by re-indexing and truncation, then there is a polynomial P with the same degree and leading coefficient as Q so that P (k)=pk.Ifpk = qk + rk then P = Q + R. The leading coefficient of P is Q0 if deg(Q) > deg(R) and is Q0 + R0 otherwise. { } { } Finally, suppose that pk is obtained from rk by integration. We will k d need the fact that i=0 i is a polynomial of degree d+1 with leading coefficient 1/(d + 1). Using the quoted fact, there is a polynomial P such that deg(P )= deg(R) + 1, the leading coefficient of P is R0/ deg(P ), and P (k)=pk. Item (3) follows easily.

(4) The statement is clear for constant sequences and if {pk} is obtained by re-indexing and truncating a sequence where the lemma holds. If {pk} is the sum of sequences {qk} and {rk} for which the statement holds and where {qk} is not constant, then {qk − c} is eventually polynomial and hence so is { − } { − } { } { }∞ pk c = qk c + rk . Finally, suppose pk k=k is obtained from the { } 0 ··· nonzero sequence qk by integration. Suppose that qk0 + qk0+1 + + qk1 >c. − ··· − ··· Then pk c is eventually (qk0 + + qk1 c)+(qk1+1 + qk1+2 + + qk). Thus, after re-indexing and truncating, we see that {pk − c} is the sum of a constant sequence and the integral of a polynomial sequence. In particular, {pk − c} is eventually polynomial.

We now record some general properties of polynomial sequences of paths that will be needed. 24 MLADEN BESTVINA, MARK FEIGHN, AND MICHAEL HANDEL

Lemma 4.6 (Stability in G). If {Ak} is a polynomial sequence of paths in G, then either {Ak} is constant or, for all N, the initial and terminal paths of Ak of length N are eventually constant.

Proof. The proof is by induction on the complexity of {Ak}. The conclusion is true if {Ak} is constant. The re-indexing and truncation step and the inversion step follow immediately from deﬁnitions. Suppose that {Bk} and {Ck} are polynomial sequences in G for which the conclusion holds. If {Ak} is obtained from {Bk} and {Ck} by concatenation, then {Ak} is constant if and only if {Bk} and {Ck} are constant. Suppose that {Bk = B} is constant, but that {Ck} is not. If C is eventually the initial path of length N of Ck, then eventually the initial path of length N of Ak is the initial path of length N of BC. Eventually, the terminal path of length N of Ak is the terminal path of length N of Ck. The case where {Bk} is not constant, but {Ck} is constant is symmetric. Finally, if neither {Bk} nor {Ck} is constant, then eventually the initial (respectively terminal) path of length N of Ak equals the initial path of length N of Bk (respectively Ck). Suppose that {Ak} is obtained from {Bk} by integration. By deﬁnition, the initial path of length N of Ak is eventually constant. If {Bk = B} is constant, then eventually the terminal path of length N of Ak is the terminal path of length N of a concatenation of B’s. If {Bk} is not constant, then eventually the terminal path of length N of Ak is the terminal path of length N of Bk.

Lemma 4.7. If {Bk} is a polynomial sequence of paths in G and if {A} and {C} are constant sequences such that eventually the terminal endpoint of A is the initial endpoint of Bk and the terminal endpoint of Bk is the initial endpoint of C, then the sequence {[ABkC]} is eventually polynomial.

Proof. The proof is by induction on the complexity of {Bk}. The statement is true if {Bk} is constant. The re-indexing and truncation step and the inversion step follow immediately from deﬁnitions. { } { } { } { } Suppose Bk = BkBk is the concatenation of Bk and Bk and the { } { } { } lemma holds for Bk and Bk .If Bk is constant, then [ABkC]=[ABkBk C]=[[ABk][Bk C]] { } and we are done by hypothesis. The case that Bk is constant is similar. { } { } If Bk and Bk are not constant then eventually [ABkBk C]=[ABk][Bk C] and again we are done by hypothesis. { } { }∞ Finally, suppose Bk is obtained from Bk k=k by integration and that { } 0 the lemma holds for Bk . Choose N so that · { } N lengthG(Bk0 ) > max lengthG(A), lengthG(C) . THE TITS ALTERNATIVE FOR Out(Fn) II 25

Then, eventually ··· [ABkC]=[ABk0 Bk0+1 BkC] ··· ··· ··· =[ABk0 Bk0+N ][Bk0+N+1 Bk−N ][Bk−N+1 BkC]. Since we have already veriﬁed the concatenation step, all three terms give polynomial sequences.

Deﬁnition 4.8. A subgroup H of a group J is primitive if, for all a ∈ J and all i =0, ai ∈ H implies that a ∈ H. An element a of J is primitive if a is a primitive subgroup of J.

Lemma 4.9. Let G → G be an immersion of finite graphs such that Im[π1(G ) → π1(G)] is a primitive finitely generated subgroup of π1(G). Let {Ak} be a polynomial sequence of paths in G. Assume that for infinitely many values of k the path Ak lifts to G starting at a given vertex x ∈ G . Then the same is true for all large k. Furthermore, the lifts form (after truncation) a polynomial sequence in G (so that in particular — see Lemma 4.6 — the terminal endpoint of these lifts is constant).

The lemma fails if the primitivity assumption is dropped; e.g. take G to be the circle and G the double cover.

Proof. We proceed by induction on the complexity of {Ak}. Suppose first that the last operation is inversion. For infinitely many k the other endpoint of the lift of Ak starting at x is a point y ∈ G (there are finitely many preimages of the common terminal endpoint of the Ak’s in G). Applying the statement of the lemma to {Ak} we learn that for all large k there is a lift Ak of Ak that terminates at y. For infinitely many k, Ak starts { } at x, and Ak forms a polynomial sequence. Therefore, for all large k, Ak starts at x. Suppose next that the last operation is concatenation: Ak = BkCk. Then Bk lifts to G starting at x for infinitely many k and thus for all large k, and the lifts Bk form a polynomial sequence. Let y be the common terminal endpoint of the Bk. Similarly, for all large k the path Ck lifts to a path Ck starting at y, and these paths form a polynomial sequence. Thus Ak = BkCk is a polynomial sequence starting at x and projecting to Ak. Finally, suppose that the last operation is integration:

Ak = B1B2 ···Bk.

Since Ak is a subpath of Al for all l ≥ k, it follows from our assumptions that each Ak lifts to a path Ak starting at x0 = x. Infinitely many of these end at the same point y1. Thus for infinitely many k the path Bk lifts starting at y1. It follows that, for all sufficiently large k, Bk lifts to a path starting at y1 26 MLADEN BESTVINA, MARK FEIGHN, AND MICHAEL HANDEL and ending at a point y2. Repeating this procedure, we produce a sequence {y1,y2, ···} such that Bk eventually lifts to a path starting at yj and ending at yj+1. Choose i

Lemma 4.10. If {Ak} is a polynomial sequence of closed paths in G based at say x, if Z is a primitive cyclic subgroup of π1(G, x), and if, for inﬁnitely many k, Ak represents an element of Z, then eventually Ak represents an element of Z.

Proof. Let (G ,x) → (G, x) be an immersion such that Im[π1(G ,x) → π1(G, x)] = Z. By Lemma 4.9, eventually Ak lifts to a closed path in G based at x.

Throughout the rest of this section, p :Γ→ G denotes the universal covering of the marked real graph G, T is a tree in in T , X is a model for T , and h : G → X a cellular homotopy equivalence such that the image of each edge of G is in normal form. By subdividing if necessary, we may assume that h−1(X(0))=G(0). In particular, the h-image of each edge of G is either horizontal or vertical. Edges of the former type are h-horizontal; edges of the latter type are h-vertical. Recall (Section 2.7) that X comes with a map q : X → T = T/Fn. By Proposition 2.5, BCC(qh :Γ→ T ) < ∞ where qh is a lift of qh. By deﬁnition, if AB is a concatenation of paths in G, then the overlap length of [h(A)] and [h(B)] is less than BCC(qh). The main technical result of this section is Proposition 4.21 which states that if {Ak} is a polynomial sequence of closed paths in G, then {T ([[Ak]])} is eventually a polynomial sequence of real numbers.

Example 4.11. Suppose that G = Rose2 with X and f as in Example 2.7. { −1 k}∞ Suppose also that the edge ‘t’ has length 1. The sequence e1e2 e1e2 k=1 is polynomial. Yet, { −1 k }∞ { ···} T ([[e1e2 e1e2]] k=1 = 0, 1, 2, 3 is eventually polynomial, but is not a polynomial sequence of nonnegative real numbers (see Lemma 4.5(1)).

Deﬁnition 4.12. A polynomial sequence {Ak} of paths in G is elliptic (with respect to h)if[h(Ak)] = νk with νk vertical. A sequence {νk} of vertical edge paths in X is elliptic if for some elliptic sequence {Ak} of edge paths in G,[h(Ak)] = νk. THE TITS ALTERNATIVE FOR Out(Fn) II 27

Lemma 4.13. The image of G(0) under h is X(0). If x, y ∈ G(0) and if P is a path in X between h(x) and h(y), then there is a unique path Q in G between x and y such that [hQ]=[P ].

Proof. We are assuming that h−1(X(0))=G(0). Since T is minimal, qh is (0) onto. Since q induces a bijection between X(0) and T , the ﬁrst statement follows. The second statement follows from the assumption that h is a homotopy equivalence.

Deﬁnition 4.14. A polynomial sequence of paths in G is short if it is a concatenation of constant and elliptic sequences.

Lemma 4.15. Let {Ak} be a short polynomial sequence in G. Then Ak is a concatenation of paths

(∗) Ak = V0,kHˆ1V1,kHˆ2 ···HˆM VM,k such that

(1) {Vi,k}k is an elliptic sequence in G,

(2) Hˆi is an h-horizontal edge of G, and

(3) eventually [h(Ak)] = h(V0,k)H1h(V1,k)H2 ···HM h(VM,k) where h(Hˆi)= Hi.

Proof. By writing elements of constant sequences as concatenations of h-vertical and h-horizontal edges, we have

Ak = V0,kHˆ1V1,kHˆ2 ···HˆM VM,k with all the desired properties except perhaps Item (3). We proceed by induction on the number of elliptic sequences in the concatenation. If there are no elliptic sequences, then {Ak} is constant and the conclusion follows. If [h(Ak)] is not eventually as in Item (4) then there is an i such that, for infinitely many k, HîVi,kHî+1 is elliptic. For these values of k, the path [h(HîVi,kHî+1)] has a common initial and terminal endpoint z. Let x be the initial endpoint of Hî and let y be the terminal endpoint of Hî+1. By Lemma 4.13, there is path σ in G connecting y to x such that [h(σ)] is the trivial path at z. By Lemma 4.7, [HîVi,kHî+1σ] is a polynomial sequence of paths. By Lemma 4.10 and the fact that edge stabilizers in T are primitive cyclic, [HîVi,kHî+1σ] and hence [HîVi,kHî+1] is eventually elliptic. Thus, ˆ ˆ Vk := Vi−1,kHiVi,kHi+1Vi+1,k is eventually elliptic. Now, ˆ ˆ ··· ˆ ˆ ··· ˆ Ak = V0,kH1V1,kH2 Hi−1VkHi+2 HM VM,k has fewer elliptic sequences in the concatenation. 28 MLADEN BESTVINA, MARK FEIGHN, AND MICHAEL HANDEL

Definition 4.16. We call the expression (∗)astable normal form for {Ak}. It is obtained by erasing pairs of h-horizontal edges Hî and Hî+1 such that [h(Vi−1,kHîVi,kHî+1Vi+1,k)] is vertical for infinitely many k (equivalently eventually vertical).

Remark 4.17. It follows from Lemma 4.15 that if {Ak} is short then the length in X of [h(Ak)] is eventually constant.

Deﬁnition 4.18. A polynomial sequence of paths {Ak} in G is long if, given N>0, there are sequences {Bk}, {Ck}, and {Dk} such that

(1) {Bk} and {Dk} are short,

(2) eventually the lengths of [h(Bk)] and [h(Dk)] are at least N, and

(3) eventually Ak = BkCkDk.

Lemma 4.19. Suppose that {Bk} is short. Then, the integral {Ak} = {B1B2 ···Bk} either is eventually long or eventually short.

Proof. Let Bk = V0,kHˆ1V1,kHˆ2 ···HˆM VM,k be a stable normal form. Sup- pose that the stable normal form for the short polynomial sequence {BkBk+1} has Nh-horizontal edges. There are two cases.

Case 1. Assume N>M. Since the stable normal form for BkBk+1 is obtained by erasing terminal h-horizontal edges from Bk and initial h-horizontal edges from Bk+1,

• eventually Bk = B1,kB2,kB3,k where {Bi,k}k are polynomial sequences,

•{Vk} := {B3,kB1,k+1} is eventually elliptic,

• the stable normal form for B2,k has a positive number of h-horizontal edges, and

• eventually

[h(BkBk+1)] = [h(B1,k)][h(B2,k)][h(Vk)][h(B2,k+1)][h(B3,k+1)].

It follows that eventually

[h(B ···B )] = [h(B )][h(B )] k k+L 1,k 2,k [h(Vk)][h(B2,k+1)][h(Vk+1)][h(B2,k+2)] ···[h(Vk+L−1)][h(B2,k+L)]

[h(B3,k+L)].

In particular, the length in X of [h(Bk ···Bk+L)] goes to inﬁnity with L. THE TITS ALTERNATIVE FOR Out(Fn) II 29

Since

B1B2 ···Bk =(B1 ···BL)(BL+1BL+2 ···Bk−L)(Bk−L+1 ···Bk), the integral of {Bk} is eventually a concatenation of three polynomial sequences. The ﬁrst and third are short. By choosing L large, the length in X of the stable normal form of the h-image of the ﬁrst and third sequences can be made arbitrarily large. It follows that this integral is eventually long.

Case 2. Assume N ≤ M. Then, at least half of the terminal h-horizontal edges of Bk are erased with initial h-horizontal edges of Bk+1 in putting BkBk+1 into normal form. Note that in this case, M is even. Indeed, otherwise the middle h-horizontal edge of Bk is erased with the middle h-horizontal edge of Bk+1; but these are the same oriented edges, a contradiction. Set

Vk := V M Hˆ M ···Vk−1,kHˆkVM,kV0,k+1Hˆ1 ···V M − Hˆ M V M . 2 ,k 2 +1 2 1,k+1 2 2 ,k+1

By assumption, {Vk} is eventually an elliptic polynomial sequence. Now

B1B2 ···Bk = V0,1Hˆ1 ···V M − Hˆ M V1V2 ···Vk−1Hˆ M V M ···HˆM VM,k. 2 1,1 2 2 +1 2 +1,k

We see that the integral of {Bk} is eventually short.

Lemma 4.20. (1) A polynomial sequence of paths {Ak} in G is either short or long.

(2) Suppose that {A1,k}, {A2,k} and {Ak} := {A1,kA2,k} are polynomial sequences of paths in G. The overlap length in X of [h(A1,k)] and [h(A2,k)] is eventually constant.

Proof. (1) The proof is by induction on the complexity of {Ak}. Con- stant sequences are short. If {Ak} is obtained by truncating and re-indexing a short (respectively long) sequence, then {Ak} is short (respectively long). The inverse of a short (respectively long) sequence is short (respectively long). Suppose {Bk} and {Ck} are short. Suppose {Dk} = {D1,kD2,kD3,k} is long with {D1,k} and {D3,k} short. Suppose {Ek} = {E1,kE2,kE3,k} is long with {E1,k} and {E3,k} short. If {Ak} = {BkCk} then {Ak} is short. If {Ak} = {DkEk} = {(D1,k)(D2,kD3,kE1,kE2,k)(E3,k)} then {Ak} is long. If {Ak} = {BkEk} = {(BkE1,k)(E2,k)(E3,k)}, then {Ak} is long. Indeed, {BkE1,k} and {E3,k} are short, and if the lengths in X of [h(E1,k)] and [h(E3,k)] are greater than N + C where C is the eventual length of [h(Bk)], then the lengths in X of [h(BkE2,k)] and [h(E3,k)] are eventually greater than N. The other case {Ak} = {DkCk} is symmetric with the case {Ak} = {BkEk}. If {Ak} is the integral of a short sequence, then the conclusion follows from Lemma 4.19. Suppose that {Ak} is the integral of a long sequence {Dk} as 30 MLADEN BESTVINA, MARK FEIGHN, AND MICHAEL HANDEL above. Suppose further that the lengths of [h(D1,k)] and [h(D3,k)] are greater than 2N if k ≥ K and that N is greater than BCC(qh). Then, eventually

Ak =(D1D2 ···DK )(DK+1DK+2 ···Dk−1D1,kD2,k)(D3,k).

Further, the length in X of [h(D1D2 ···DK )] is at least N. It follows that {Ak} is long.

(2) There are four cases depending on whether or not {A1,k} or {A2,k} is short or long. Let {Bk}, {Ck}, {Dk}, and {Ek} be as in the proof of (1) above. If both {A1,k} and {A2,k} are short, then so is {Ak}, and the conclusion follows from Lemma 4.15. Choose N to be longer than BCC(qh). Then, the overlap length of [h(A1,k)] and [h(A2,k)] is less than N. Suppose that {A1,k} = {Dk} and {A2,k} = {Ek}. Suppose that the lengths in X of [h(D3,k)] and [h(E1,k)] are eventually greater than N. Then, the overlap length of [h(A1,k)] and [h(A2,k)] is eventually the same as the overlap length of [h(D3,k)] and [h(E1,k)]. Thus, we are reduced to the case of short sequences. The other two cases, {A1,k} = {Bk}, {A2,k} = {Ek} and {A1,k} = {Dk}, {A2,k} = {Ck}, similarly reduce to the case of short sequences.

Proposition 4.21. If {Ak} is a polynomial sequence of paths (respectively { } closed paths) in G, then the sequence lengthX ([h(Ak)]) (respectively {T ([[Ak)]])}) of real numbers is eventually polynomial. If P is a polynomial { } such that eventually P (k)=lengthX ([h(Ak)]) (respectively T ([[Ak]]) ) then the leading coeﬃcient of P is bounded below by m/ deg(P )! where m is the length of the shortest edge of T .

Proof. We prove the statements about sequences of paths; the case of closed paths is similar and is left to the reader. The proof is by induction on the { } { } complexity of Ak with the induction statement being that lengthX ([h(Ak)]) is eventually polynomial and that the positive constant sequences of real numbers used in the integration operations are realized as lengths of edge paths in T . This directly proves the first statement of the proposition and the second follows from Lemma 4.5(3b). The case that {Ak} has zero complexity follows from the first item in Remark 0.1. We now assume the desired result for all polynomial paths with complexity less than that of {Ak}. By Lemma 4.20(1), {Ak} either is short or long. In the former case, the induction statement follows from Remark 4.17 and the first item in Remark 0.1. We now assume that {Ak} is long. If {Ak} is obtained from a sequence satisfying the inductive statement by inversion or by truncation and re-indexing, then {Ak} satisfies the inductive statement. THE TITS ALTERNATIVE FOR Out(Fn) II 31

Suppose that {Ak} is the concatenation of {A1,k} and {A2,k} where both {A1,k} and {A2,k} satisfy the inductive statement, and at least one of {A1,k} and {A2,k} is not short. Let A be the eventual overlap length of [h(A1,k)] and [h(A2,k)] (see Lemma 4.20(2)). Then, eventually the length in X of [h(Ak)] equals − lengthX ([h(A1,k)]) + lengthX ([h(A2,k)]) 2A. { } Lemma 4.5(4) implies that lengthX ([h(Ak)]) is eventually polynomial and hence that the inductive statement is satisﬁed. Finally, suppose that {Ak} is the integral of the sequence {Bk} where {Bk} satisﬁes the induction statement. By Lemma 4.20(2), the overlap length of [h(Bk)] and [h(Bk+1)] is eventually a constant B. After re-indexing and { } { } { − truncating, lengthX ([h(Ak)]) is 2B plus the integral of lengthX ([h(Bk)]) 2B}. The latter sequence is polynomial by Lemma 4.5(4) and is eventually the length of an edge path in T by the second item in Remark 0.1 with D = D = B. This completes the induction step and so also the proof of the proposition. ∈T O∈ ∈ Let T , UPGFn , and a Fn. By Lemma 4.2 and Proposition 4.21, k the sequence {T (O ([[a]]))} is eventually polynomial. The degree d(O,T,a) of the polynomial is uniformly bounded by the number of strata in a UR for O. Let d(O,T) = max{d(O,T,a) | a ∈ Fn}.

Theorem 4.22. Suppose T ∈T and O∈UPG(Fn). Set d = d(O,T). Then, the sequence {T Ok/kd} converges to a tree T O∞ ∈T.

Proof. By Lemma 4.2 and Proposition 4.21 and the deﬁnition of d, the sequences k d {T (O ([[a]]))/k } converge for all a ∈ Fn and not all of these limits are 0. Hence, the sequence k k {T O /d } converges to a nontrivial Fn-tree. That this tree is very small follows from the fact, proved in [CL95], that TVS ∪{trivial action} is closed under limits. By Lemma 4.5(3), if P is a polynomial of degree d such that eventually k P (k)=T (O ([[a]])), then the leading coeﬃcient of P is bounded below by a nonzero constant depending only on d and the lengths of the edges of T .It k d follows that the collection of nonzero numbers of the form lim T (O ([[a]]))/k are bounded away from 0, and so T O∞ is simplicial.

Question. Does Theorem 4.22 hold for trees in TVS?

Deﬁnition 4.23. Trees T for which d(O,T) = 0, i.e. for which the sequence { Ok }∞ T ( ([[γ]])) k=0 is eventually constant for every γ ∈ Fn, are O-nongrowers. Others are O- growers.Ifd(O,T) = 1, then we say that T grows linearly under O. 32 MLADEN BESTVINA, MARK FEIGHN, AND MICHAEL HANDEL

Remark 4.24. There exist nongrowers that are not fixed. An example is the F2-tree T pictured below. The first map Rose2 → X is the marking and is given by identifying the two bold arcs. (Recall that the edges of Rose2 are identified with the elements of the standard basis {e1,e2}.) All edges of X are horizontal except for the image e1. The tree T is O-nongrowing where O is represented by the automorphism given by e1 → e1, e2 → e1e2. Such examples do not exist in SLn(Z) or the mapping class group of a surface.

Rose2 X T

Nongrowers are also responsible for the existence of automorphisms O and compact sets K ⊂ PTVS in the complement of Fix(O) with the property that for no k is KOk contained in a prescribed small neighborhood of Fix(O). Here PTVS is the space of homothety classes of very small Fn-trees. A concrete example can be described as follows. Let O∈Out(F4) be represented by the automorphism given by e1 → e1 and ei → eie1 if i>1. The compact set K consists of the bi-infinite sequence {...,T−2,T−1,T0,T1,T2,...} together with the limiting tree T∞. The tree T k is pictured below. Here we have indicated the marking by the elements of F4 on the closed edges (i.e. loops) of X. Only the edge marked e1 is vertical. Passing to the limit as k goes to infinity amounts to opening up the loops of T which in the limit correspond to e2 and e3.Now k notice that TkO converges to a nonfixed nongrower (which is a tree just like T∞ except for a permutation of {e2,e3,e4}). It is, however, true that if K is a compact subset of PTVS consisting of growers, then the accumulation set of the sequence KOk is a subset of Fix(O).

4.2. Explicit limits. The goal of this section is to give an explicit construction of T O∞ for T ∈T in the case that d = d(O,T) ≥ 1, an assumption we make for the rest of this section. The case that O has linear growth was done by Cohen and Lustig in [CL95]. Let f : G → G be a UR representing O. Unless otherwise stated, paths in graphs are assumed to begin and end at vertices.

Notation 4.25. For a path σ in G, f#(σ) is the path [f(σ)]. THE TITS ALTERNATIVE FOR Out(Fn) II 33

k e2e1 q

e −k 1 e3e1

T k X

By Theorem 3.11(5), a suﬃciently high iterate of a path σ has a splitting into exceptional subpaths and single edges. The following lemma implies that σ itself has a decomposition (not necessarily a splitting) of this sort.

Lemma 4.26. Maximal exceptional subpaths of a path do not overlap.

k Proof. Suppose that σ1 = Eiτ E¯j is an exceptional subpath of σ and that σ2 is an exceptional subpath of σ that intersects σ1 in a proper nontrivial subpath of σ1. Either the initial edge of σ2 is contained in σ1 and is not Ei or the terminal edge of σ2 is contained in σ1 and is not E¯j. These two cases are interchanged when σ is replaced byσ ¯ so that there is no loss in assuming that the former holds. The heights of the ﬁrst and last edges of an exceptional path are strictly greater than the height of any other edge in the exceptional path. Thus if σ2 contains E¯j then E¯j is the ﬁrst or last edge of σ2. Since no exceptional path begins with E¯j, σ2 cannot extend past E¯j and σ2 is a subpath of σ1.

Lemma 4.26 implies that there is a well deﬁned decomposition σ = σ1 ...σm into single edges and maximal exceptional subpaths. We call this the canonical decomposition of σ.

Deﬁnition 4.27. Let h : G → X be a homotopy equivalence of G to a model for T taking vertices to vertices. For any path σ ⊂ G with endpoints at vertices let d(σ)=d(σ, X) be the degree of the polynomial sequence k lengthX ([h(f (σ))]), and let k d LG(σ) = lim lengthX ([h(f (σ))])/k . k→∞

The notation LG is chosen to remind the reader that LG is a length function on paths in G. Proposition 4.21 implies that LG(σ) is well deﬁned. By deﬁnition, 34 MLADEN BESTVINA, MARK FEIGHN, AND MICHAEL HANDEL

LG([f(σ)]) = LG(σ) for all σ. Theorem 4.22 implies that LG agrees with the length function induced by T O∞ on circuits in G. The problem with this definition of LG is that it requires iteration and ⊂ ∗ m taking limits. For any path σ G, let LG(σ)= i=0 LG(σi) where σ = σ1 ...σm is the canonical decomposition of σ. We show below (Proposition ∗ 4.32) that LG = LG. Along with Lemma 4.28, this allows us to compute LG directly once its value on a finite number of paths is known. We say that an edge Ei is linear if its suffix ui is a Nielsen path; in this ni case we write ui = τi for a primitive Nielsen path τi and some ni > 0. If τ is a primitive Nielsen path and if there is more than one linear edge whose suffix is an iterate of τ then we say that {Ei : τi = τ} is the linear family associated to τ. By reordering the edges of G we may assume that the edges in a linear family are consecutively numbered, say {Ei : s ≤ i ≤ t}. The following lemma gives LG(σ) when σ is a linear edge or an exceptional path.

Lemma 4.28. Suppose that τ is a primitive Nielsen path, that Ei and Ek m belong to the linear family associated to τ and that σ = Eiτ E¯k. Then

(1) If T ([[τ]]) = 0 then d(Ei)=d(Ek)=d(σ)=0and LG(Ei)=LG(Ek)= LG(σ)=0.

(2) If T ([[τ]]) > 0 then d(Ei)=d(Ek)=1and d(σ) ≤ 1. In particular, if d>1, then LG(Ei)=LG(Ek)=LG(σ)=0.

(3) If T ([[τ]]) > 0 and d =1,then LG(Ei)=ni · T ([[τ]]),LG(Ek)=nk · T ([[τ]]) and LG(σ)=T ([[τ]]) ·|ni − nk|.

Proof. This is a direct consequence of the bounded cancellation lemma m and the fact that for any a ∈ Fn and m>0, T ([[a ]]) = mT ([[a]]).

Lemma 4.29. If σ has a splitting σ = σ1 ···σm into exceptional subpaths and single edges, then σ = σ1 ···σm is the canonical decomposition of σ and ∗ LG(σ)=LG(σ). Proof. Lemma 4.26 implies that if τ is a maximal exceptional subpath of σ then τ = σj ...σk for some 1 ≤ j ≤ k ≤ m. Since exceptional paths have no nontrivial splittings, j = k and τ = σj. This proves that σ = σ1 ···σm is the canonical decomposition of σ. Lemma 4.20(2) implies that there is a uniform k k bound between lengthX ([h(f (σ))]) and lengthX ([h(f (σi))]). The equality ∗ ≥ LG(σ)=LG(σ) therefore follows from the assumption that d 1.

We say that an edge Ei is below the edge Ej, i = j, if the canonical decomposition of either f(Ej)orf(E¯j) contains a term that is not a Nielsen path and that has Ei as its initial edge. If Ei is not below any edge then it is topmost. THE TITS ALTERNATIVE FOR Out(Fn) II 35

Lemma 4.30. ∗ LG(σ)=LG(σ)=0if σ is any of the following: • a suﬃx,

• an edge that is not topmost,

• an edge that is ﬁxed by f,

• a Nielsen path.

Proof. Suppose that u is the suffix of an edge E. Remark 3.14 and Theorem 3.11(2) imply that u has a splitting u = u1 ···um into exceptional subpaths and single edges and that f(E) has a splitting f(E)=E · u1 ···um. Lemma 4.29 and the fact that LG([f(E)]) = LG(E) therefore imply the first ∗ ∗ item and that each LG(ui)=LG(ui) = 0. Since LG and LG agree on edges, to prove the second and third items we need only prove that LG(E) = 0 when E is either not topmost or is fixed. The former case follows from Lemma 4.28 applied to a ui as above and the latter from the assumption that d ≥ 1. If σ is a Nielsen path then σ =[f k(σ)] for all k so that σ has a splitting ∗ into exceptional subpaths and single edges. Lemma 4.29 implies that LG(σ)= LG(σ). Since d ≥ 1, LG(σ)=0.

Lemma 4.31. ∗ Assume that LG(α)=0. • ∗ If the initial endpoint of σ equals the terminal endpoint of α then LG([ασ]) ∗ = LG(σ). • ∗ If the terminal endpoint of σ equals the initial endpoint of α then LG([σα]) ∗ = LG(σ). ∗ ∗ Proof. Since LG(ρ)=LG(¯ρ) for all ρ, the two items are equivalent. We prove the ﬁrst item by induction on the number of terms in the canonical decomposition of α. Let σ = σ1 ...σm be the canonical decomposition of σ.Ifα is a single edge there are four cases to consider. If ασ1 ...σm ∗ ∗ is the canonical decomposition of [ασ] then LG([ασ]) = LG(σ)+LG(α)= ∗ LG(σ). If σ1 =¯α then σ2 ...σm is the canonical decomposition of [ασ] and ∗ ∗ − ∗ ¯ m ¯ so LG([ασ]) = LG(σ) LG(α)=LG(σ). If α = Ei and σ1 = Eiτ Ek is an exceptional path, then the canonical decomposition of [ασ] is the canonical m decomposition of the Nielsen path τ followed by E¯k followed by σ2 ...σm. ∗ m By Lemma 4.28, LG(σ1)=LG(Ek) = 0. By Lemma 4.30, LG(τ )=0.Thus ∗ ∗ − ∗ m ∗ LG([ασ]) = LG(σ) LG(σ1)+LG(τ )+LG(Ek)=LG(σ). The remaining case m m is that α = Ei, that σ begins with τ E¯k and that Eiτ E¯k is an exceptional ∗ ∗ ∗ path. Then LG(σ)=LG([¯α[ασ]]) = LG([ασ]) by the previous case. This completes the proof when α is a single edge. 36 MLADEN BESTVINA, MARK FEIGHN, AND MICHAEL HANDEL

m If α = Eiτ E¯k is exceptional, then there are three cases. By Lemma 4.28, ∗ m LG(Ei)=LG(Ek). By Lemma 4.30, LG(τ ) = 0. If the initial edge of σ is not ∗ ∗ ∗ ∗ ∗ Ek then LG([ασ]) = LG(α)+LG(σ)=LG(σ). If σ1 = Ek then LG([ασ]) = ∗ − ∗ m ∗ p ¯ LG(σ) LG(Ek)+LG(τ )+LG(Ei)=LG(σ). Finally, if σ1 = Ekτ El is m+p exceptional then the canonical decomposition of [ασ]is(Eiτ E¯l)σ2 ...σm ∗ ∗ − m ¯ ∗ m+p ¯ ∗ so LG([ασ]) = LG(σ) LG(Eiτ Ek)+LG(Eiτ El)=LG(σ). This completes the proof when there is only one term in the canonical decomposition of α. In general, let α1 be the ﬁrst term in the canonical decomposition and ∗ ∗ ∗ ∗ write α = α1α . Then LG([ασ]) = LG([α1[α σ]]) = LG([α σ]) = LG(σ) where the last equality is by induction.

Proposition 4.32. ⊂ ∗ For any path σ G, LG(σ)=LG(σ).

k ∗ k Proof. By Theorem 3.11(5), LG(σ)=LG([f (σ)]) = LG([f (σ)]) for all ∗ sufficiently large k. It therefore suffices so show that LG is f#-invariant. We argue by induction on the height h(σ)ofσ.Forh(σ) = 0 the statement ∗ ∗ is vacuously true. Assume that height(σ)=h and that LG([f(σ)]) = LG(σ) for paths σ with height at most h − 1. If Eh is part of a linear family, denote this family {Ei : s ≤ i ≤ t}; otherwise let s = h. The edges Ei with s ≤ i ≤ h are all highest edges so that there is a splitting (see Definition 3.15) of σ into subpaths of the following types: µ, Eiµ, µE¯i and EiµE¯j where s ≤ i, j ≤ h and where µ is contained in Gs−1. The canonical decomposition of σ is a refinement of this splitting and so it suffices to assume that σ has one of these forms. The case that σ = µ follows by induction. Suppose that σ = Eiµ. Since f is a homotopy equivalence and fixes all vertices, there is a path ηi of height at most s − 1 such that f#(ηi) is the suffix ui of Ei. By Lemma 4.31 and ∗ ∗ the inductive hypothesis, LG(ηi)=LG(ui) = 0. Since µ and [uif(µ)] are contained in Gs−1, Ei is the first term in the canonical decompositions of ∗ ∗ σ = Eiµ and [f(σ)] = Ei[uif(µ)]. Thus LG(f#(σ)) = LG(f#([[Eiη¯i][ηiµ]])) = ∗ ∗ ∗ LG(Eif#([ηiµ])) = LG(Ei)+LG(f#([ηiµ])) = LG(Ei)+LG([ηiµ]) = LG(Ei)+ ∗ ∗ ¯ ¯ LG(µ)=LG(σ). The µEi case follows by symmetry. If σ = EiµEj is exceptional, then so is [f(σ)] and Lemma 4.28 completes the proof. Otherwise both Ei and E¯j are terms in the canonical decomposition of both σ and [f(σ)]. In this case, one writes σ =[[Ehη¯i][ηiµη¯j][ηjE¯h]] and argues as in the σ = Ehµ case.

Our remaining task is to explicitly construct a tree whose length function ∗ agrees with LG.

Lemma 4.33. Let Gˆ be the subgraph consisting of edges E with LG(E)=0. If d ≥ 2, then T O∞ is the tree obtained from G by collapsing the components of Gˆ to points and assigning each remaining edge E the length LG(E). THE TITS ALTERNATIVE FOR Out(Fn) II 37

Proof. By Lemma 4.28 each exceptional subpath of σ is contained in Gˆ. One may therefore compute L∗(σ) by writing σ as a concatenation of edges and adding up the LG-length of each edge.

For the remainder of the section we assume that d = 1. To each edge E of G, assign a formal length LG(E). By reordering the edges of G we may assume that the edges in a linear family with positive formal length are consecutively numbered, say {Ei : s ≤ i ≤ t}, and that within such a linear family, formal lengths are increasing; equivalently, nj >ni if s ≤ iL(EjEi) by Lemma 4.28. To create a graph G that assigns the correct lengths to exceptional paths of positive formal length, we fold edges within linear families {Ei : s ≤ i ≤ t} of positive formal length as follows. ¯ ¯ First fold part of Et over all of Et−1. Denote the unfolded part of Et by Et and assign it a formal length equal to LG(Et) − LG(Et−1). Next fold part of ¯ ¯ Et−1 over all of Et−2 denoting the unfolded part of Et−1 by Et−1 and assign it a formal length equal to LG(Et−1) − LG(Et−2). Continue this down the linear family with the last fold deﬁning Es+1. Repeat this for all linear families with positive formal length. An edge E ⊂ G that is not subdivided in this process determines an edge in G that we label E and assign a formal length equal to LG(E). Let g : G → G be the total folding map. If Ei is a member of a linear { ≤ ≤ } family with positive length Ei : s i t then g(Ei)=EiEi−1 ...Es.We will not use f to induce a homotopy equivalence of G but simply use G as the basis for the construction of T O∞. For each linear edge Ei ⊂ G, let ηi = EiτiE¯i be the primitive Nielsen path determined by Ei. Denote [g(ηi)] by ηi and let τi be the subpath satisfying ¯ ηi = Eiτi Ei.Ifg(Ei) is a single edge Ei, then τi =[g(τi)]. Otherwise Ei is an element of an exceptional family {Ei : s ≤ i ≤ t} and i>s. In that case ¯ ¯ τi = Ei−1 ...EsτiEs ...Ei−1 = ηi−1. Remark 4.34. By Theorem 3.11(5), a path σ ⊂ G is a Nielsen path if and only if it is a concatenation of subpaths, each of which is either a ﬁxed edge k k ¯ or [ηi ]=Eiτi Ei for some integer k.

Lemma 4.35. Suppose that σ is a path in G and that σ = σ1 ...σm is its canonical decomposition. Denote [g(σ)] by σ and [g(σi)] by σi. Then • σ = σ1 ...σm is a decomposition into subpaths. • { } m The maximal subpaths αk of σ of the form ηl are disjoint. • Each αk is contained in some σj. • For each j, the sum of the formal lengths of the edges of σj that are not contained in any αk equals LG(σj). 38 MLADEN BESTVINA, MARK FEIGHN, AND MICHAEL HANDEL

Proof. We begin by enumerating the possible values of σj.Ifσj is a single { ≤ ≤ } edge Ei then σj = Ei unless Ei is contained in a linear family Ej : s j t m ¯ ≤ ≤ and i>s. In that case, σj = EiEi−1 ...Es.Ifσj = Eiτs Ek where s k ≤ m ¯ ¯ m i t then σj = EiEi−1 ...Ek+1Ek ...Esτi Es ...Ek = EiEi−1 ...Ek+1[ηk ]. Given these possibilities, the third item implies the second and fourth items. If the first item fails then someσ ¯j and σj+1 have a common initial edge. Examining the values above and keeping in mind that the initial edges ofσ ¯j and σj+1 are distinct, we see that the only possibility is that σj = Ei and σj+1 begins with E¯s for some linear family {Ej : s ≤ j ≤ t} with s ≤ i ≤ t. But then there is an exceptional subpath that overlaps with σj and σj+1 in contradiction to the definition of canonical subpath. This proves the first item. The homotopy equivalence g induces a bijection on paths with endpoints at vertices. Given a subpath µ ⊂ σ let µ ⊂ σ be the smallest path (not necessarily with endpoints at vertices) satisfying [g(µ)] = µ. Suppose that m µ =[ηl ]. If El is not part of a linear family then µ must have endpoints m at vertices and so, by uniqueness, must be [ηl ]. In particular, µ is contained in some σj and so µ is contained in some σj.IfEl is part of a linear family then it may be that one or both endpoints of µ are contained in the interior of edges in the same linear family as El. In this case the path ν obtained from µ by including the edges that contain the endpoints of µ has image of the form ¯ m ¯ Ejτs Ek. By uniqueness ν is exceptional and we conclude as before that µ is contained in a single σj. This proves the third item and so also the second and fourth items.

Definition 4.36. Let Gˆ be the subgraph of G consisting of edges with formal length zero and let T 0 be the tree determined from G by collapsing the components of Gˆ to points and making the length of each remaining edge equal to its formal length. For i>0 we will inductively define T i working our way up the strata of G.IfEi is not a linear edge with positive formal length, i i−1 then g(Ei)=Ei is a single edge and we define T = T . Otherwise, define i i−1 T to be the tree obtained from T by pulling τi over Ei [BF91, p. 452]. i−1 Remark 4.34 implies that τi is elliptic in T and so this operation is well defined. Denote the tree obtained at the end of the process by T .

Lemma 4.37. If d =1then T O∞ = T .

{ } Proof. Given a circuit σ contained in G, denote [g(σ)] by σ and let αk be as in Lemma 4.35. By Lemma 4.35, it suﬃces to show that T ([[σ ]]) equals the sum of the formal lengths of the edges of σ that are not contained in any αk. But this is exactly how T was constructed. THE TITS ALTERNATIVE FOR Out(Fn) II 39

Remark 4.38. Nontrivial edge stabilizers of T O∞ all arise from Deﬁni- tion 4.36. We record for future use the fact that each nontrivial edge stabilizer of T is generated by a conjugate of a root of a linear suﬃx of f.

4.3. Primitive subgroups. From the homology, it is clear that if the elements of a free factor system are permuted under a unipotent outer automorphism, then each is invariant, and the restriction is unipotent. We will show moreover that a periodic free factor (or even a vertex stabilizer of a tree in TVS) is invariant. Our argument uses only that vertex stabilizers are primitive. Recall that a subgroup H of a group J is primitive if, for all a ∈ J and all i =0, ai ∈ H implies that a ∈ H. An element a of J is primitive if a is a primitive subgroup of J.

Lemma 4.39. Let H be a ﬁnitely generated nontrivial primitive subgroup of Fn. Then the normalizer N(H) of H in Fn is H.

Proof. Let T be a minimal free simplicial Fn-tree and let TH be a minimal H-invariant subtree of T . Let γ ∈ N(H). Then γ(TH )=TH and so the axis of γ is in TH and projects to a circuit in TH /H. Thus, a power of γ is in H. Since H is primitive, γ is in H.

Lemma 4.40. Let H be a subgroup of Out(Fn) and let H be a ﬁnitely generated primitive subgroup of Fn whose conjugacy class is H-invariant. Then, the restriction map ρH : H→Out(H) is well-deﬁned. Further, if H⊂PG(Fn), then H|H := ρH (H) ⊂ PG(H).

Proof. The ﬁrst statement is an easy consequence of Lemma 4.39. The second statement follows from the fact that the inclusion of a ﬁnitely generated subgroup into Fn is a quasi-isometry, see for example [Sho91].

Proposition 4.41. Suppose that O∈UPG(Fn) and that H ⊆ Fn is a primitive ﬁnitely generated subgroup. If Ok([[H]])=[[H]] for some k>0, then k O([[H]]) = [[H]]. Furthermore, if Φ ∈ Aut(Fn) is a lift of O with Φ (H)=H then Φ(H)=H.

The statement is false without the primitivity assumption as the following 2 example shows: F2 = e1,e2 ,Φ(e1)=e1,Φ(e2)=e1e2, H = e1,e2 , k =2. Proof. By Proposition 3.16, we may assume that rank(H) > 1. Let f : G → G beaURforO such that f is linear on each edge of G.By p : G → G denote the covering space of G corresponding to H. There is a lift g : G → G of f k. We claim that, after replacing g by a power if necessary, g fixes a vertex. Indeed, by linear algebra, some power gm of g will have negative Lefschetz number. Any fixed point v of negative index of gm composed with the retraction to the core is fixed under gm. The claim now follows from the 40 MLADEN BESTVINA, MARK FEIGHN, AND MICHAEL HANDEL observations that p(v) is a fixed point for a positive power of f, that a positive power of a UR is also a UR, and that the only fixed points of a UR that is linear on edges are vertices and fixed edges. We now use v and p(v) as base points. Let σ be a closed path in G based at v. The sequence {[gi(σ)]} forms a sequence of lifts of a subse- quence of the sequence {[f j(p(σ))]}. The latter is eventually a polynomial sequence (Lemma 4.2), and hence by Lemma 4.9 eventually the based closed path [f j(p(σ))] lifts to a based closed path in G. Applying this to closed j paths σ representing generators of π1(G ,v) we conclude that f eventually lifts to G. Thus, eventually Oj([[H]])=[[H]], and the claim follows. For the “furthermore” part of the proposition let v be the base point and choose g so that v is fixed.

Deﬁnition 4.42. A circuit in a graph is primitive if it is not a proper power of another circuit.

Lemma 4.43. Let G be a ﬁnite connected core graph with oriented edges. Suppose that f : G → G is a cellular homeomorphism preserving orientations and inducing a nontrivial permutation of the edges. Then either

(1) there is a primitive circuit σ = E1E2 ···Em that is nontrivially rotated by f, i.e. there is a p,0

(2) there is an f-ﬁxed vertex such that f nontrivially permutes the edges containing v.

Proof. First suppose that v, f(v),...,fm(v)=v is a nontrivial orbit of vertices with m minimal. Choose an embedded path τ connecting v and f(v). If τf(τ) ···f m(τ) is essential, then [[τf(τ) ···f m(τ)]] is a circuit as in (1). If not, then a vertex in τ is as in (2). If f ﬁxes the vertices of G, then the common vertex of a nontrivial orbit of edges is as in (2).

Proposition 4.44. Suppose that O∈UPG(Fn) and that H ⊆ Fn is a primitive ﬁnitely generated subgroup whose conjugacy class is ﬁxed by O. Then the restriction (see Lemma 4.40) O|H ∈ UPG(H).

Proof. It is obvious that O|H ∈ PG(H). We argue that the action on homology is unipotent. Let f : G → G beaURforO.Byp : G → G denote the covering space of G corresponding to H and let f : G → G be a lift of f. By C denote the core of G. Let ρ : G → C be the nearest point retraction. If C does not contain any lifts of the highest edge E ⊂ G, then we may argue by induction on the number of strata. Therefore we assume that C contains lifts THE TITS ALTERNATIVE FOR Out(Fn) II 41 of E. From C form a finite graph Cˆ by collapsing all edges that do not project to E. The map ρf induces a cellular isomorphism fˆ : Cˆ → Cˆ. Since C is finite, fˆ has finite order. The main step of the proof is to argue that fˆ = id. If fˆ is not the identity, then, according to Lemma 4.43, there are two cases. Suppose first that there is a primitive circuit σ = E1E2 ···Em that is nontrivially rotated by fˆ. Here each Ei is a lift of E or of E, and, say, fˆ(Ei)=Ep+i,0

4.4. Unipotent automorphisms and trees. See Section 2 for deﬁnitions.

Proposition 4.45. Let O∈UPG(Fn) and T ∈TVS. Suppose that, Point(T ) and Arc(T ) are O-invariant. Then, (1) each element of Arc(T ) is O-invariant, (2) each element of Point(T ) is O-invariant, and

(3) if [[V ]] ∈ Point(T ), then O|V ∈ UPG(V ).

Proof. By Theorem 2.3, Arc(T ) is a ﬁnite set. Item (1) follows by Propo- sition 3.16. Similarly, each of the ﬁnitely many elements of Point(T )isO-periodic, and hence is O-invariant by Proposition 4.41. The restriction of O is unipotent by Proposition 4.44. 42 MLADEN BESTVINA, MARK FEIGHN, AND MICHAEL HANDEL

Remark 4.46. If T ∈TVS is ﬁxed by O∈Out(Fn), then the hypotheses of Proposition 4.45 are satisﬁed.

Lemma 4.47. Let f : G → G be a cellular homeomorphism of a connected ﬁnite graph. Suppose that f ﬁxes all valence one vertices and induces a unipotent map of H1(G; Z). Then either f is homotopic rel vertices to the identity or there is a homeomorphism G to S1 that conjugates f to a rotation.

Proof. After a homotopy relative to vertices, we may assume that every point is periodic. If G is a circle, the claim is clear. We may therefore assume that either G has valence one vertices or negative Euler characteristic. In either case Fix(f) = ∅. We will assume that Fix(f) = G and derive a contradiction. After replacing f by an iterate if necessary, we may assume that all nonfixed points have the same period, say k>1. Choose a shortest arc α in G that intersects Fix(f) exactly in its (possibly equal) endpoints ∂α. Suppose that there exists v ∈ α \ ∂α such that f(v) ∈ α. Since v is not a fixed point, either v or f(v) is not the midpoint of α. But then there is path in α ∪ f(α) that intersects Fix(f) exactly in its boundary and that is shorter than α.We conclude that α, f(α),...fk−1(α) intersect only in their endpoints. It follows that there is a nonfixed periodic class under the action of f k on the homology ∪k−1 i of the subgraph i=0 f (α). This contradicts Corollary 3.3.

Proposition 4.48. Assume n>1. Suppose O∈UPG(Fn) ﬁxes T ∈T. Let Φ ∈ Aut(Fn) be a lift of O and let fΦ : T → T be a Φ-equivariant isometry. Then fΦ ﬁxes all orbits of vertices and directions.

Proof. The map fΦ induces a periodic homeomorphism f Φ of the quotient graph T = T/Fn.Ifx ∈ T is a vertex such that FixT (StabT (x)) = {x}, then the image of x in T is fΦ-ﬁxed by Proposition 4.45. In particular, f Φ ﬁxes all valence 1 vertices. Since the induced action on homology of T is unipotent, by Lemma 4.47, f Φ either is the identity or is conjugate to a rotation of the circle. The latter is impossible since Elliptic(T ) = {1}.

Lemma 4.49. Let O∈UPG(Fn), S ∈TVS, and a ∈ Fn. Suppose K SO∞ (a) > 0. Then there is a K such that S(O (a)) > 0 for all K >K.

Proof. This is immediate by the deﬁnition of limits.

Proposition 4.50. Let f : G → G be a UR for O∈UPG(Fn) and let ∞ T ∈TVS be O-nongrowing. If a ∈ Fn is elliptic in T O , then a is elliptic in T .

Proof. We ﬁrst comment that, by the deﬁnition of nongrowing, if x ∈ Fn is elliptic in T O∞ then there is k(x) such that Ok([[x]]) is elliptic in T for k>k(x). THE TITS ALTERNATIVE FOR Out(Fn) II 43

∞ Let V = a1,...,ap be a point stabilizer of T O containing a. By the above comment, there is a k such that, for all pairwise products aiaj, we have k k O ([[aiaj]]) is elliptic in T . By Theorem 2.1, O ([[V ]]) is elliptic in T . Since V is a point stabilizer of the O-ﬁxed tree T O∞, O([[V ]]) = [[V ]] by Theorem 2.3 and Proposition 4.41. So [[V ]], and hence a, is elliptic in T .

5. A Kolchin theorem for unipotent automorphisms

The rest of the paper is devoted to the proof of our main theorem.

Theorem 5.1. For every ﬁnitely generated unipotent subgroup H of Out(Fn), there is a tree in T with all edge stabilizers trivial that is ﬁxed by all elements of H. Furthermore, there is such a tree with exactly one orbit of edges and if F is any maximal proper H-invariant free factor system then T can be chosen so that F(T )=F.

We ﬁx notation as follows. Let F be a maximal H-invariant proper free factor system. Choose a generating set

H = O1, O2,...,Ok

l l and for 1 ≤ l ≤ k let fl : G → G be a UR for Ol with a ﬁltration element l F l F ∈T Gr(l) such that (Gr(l))= . Choose T0 with trivial edge stabilizers and with F(T0)=F.

Deﬁnition 5.2. The bouncing sequence associated with the above choices is the sequence of simplicial trees

{T0,T1,T2, ···} in T deﬁned by O∞ Ti = Ti−1 i where subscripts of the Oi’s are taken mod k (see Theorem 4.22).

We prove Theorem 5.1 by showing that the bouncing sequence is eventually constant and that the limit tree satisfies the conclusions of the theorem. As intermediate steps, we show that Ti is Oi+1 nongrowing for all sufficiently large i (Proposition 5.7) and that the edge stabilizers of Ti are trivial for all sufficiently large i (Lemma 5.10). It turns out that the bouncing sequence is constant. This is reflected, for example, in Lemma 5.10 where having established properties of Ti for large i, the properties then hold for all i. Before starting the proof, we give some examples of what can happen if F is not assumed to be maximal. 44 MLADEN BESTVINA, MARK FEIGHN, AND MICHAEL HANDEL

Example 5.3. Recall that F2 = e1,e2. Set H = O with O represented by the automorphism Φ given by Φ(e1)=e1,Φ(e2)=e2e1, and let T0 ∈TCV . ∞ Then, T0O is the tree T from Example 2.7. The tree T is O-ﬁxed, and so the bouncing sequence is eventually constant. However, T has nontrivial edge stabilizers, and in this case the iteration scheme fails to discover a tree as in the conclusion of Theorem 5.1. Here F is trivial. We can discover a larger invariant proper free factor system, namely {[[e1]]}, by looking at the edge stabilizer.

Example 5.4. Recall F3 = e1,e2,e3. Set H = O1, O2, where Oi is represented by the automorphism Φi given by Φ1(e1)=e1,Φ1(e2)=e2e1, −1 Φ1(e3)=e3,Φ2(e1)=e1,Φ2(e2)=e2, and Φ2(e3)=e2 e1e2e3. Notice that the basis {e1,e2,e2e3} is better adapted to Φ2 since Φ2(e2e3)=e1e2e3. Consider the trees in T whose models are pictured below.

e2 O∞ 1 e2

e1 X(T0)

e1 X(T1)

e2e3 O∞ 2

e2 e3

e1 X(T2) e2 O∞ 1

e2e3

In the illustration, group elements indicate the image of the markings, and the only vertical edges are the ones corresponding to e1. All edge stabilizers are trivial and in each case the indicated tree has inﬁnite cyclic vertex sta- O∞ bilizers (always conjugate to e1 ). The tree T1 is the limit T0 1 , T2 is the O∞ limit T1 2 , etc. The tree T2 is combinatorially isomorphic to T0, i.e. T0 and T2 belong to the same open cone of T . Notice, however, that T0 and T2 are not THE TITS ALTERNATIVE FOR Out(Fn) II 45 homothetic: the ratio (e2)/(e2e3) is smaller in T2 than in T0. The bouncing sequence indeed bounces between two open cones in T , so it does not stabilize. All trees in the sequence are nongrowers under all elements of H. The ratios F { } Ti (e2)/Ti (e2e3) converge to 0. Here is [[ e1 ]] . We see that the edge marked e2 gets relatively shorter and shorter in the bouncing sequence. This tells us how to enlarge F to a larger H-invariant free factor system, namely {[[e1,e2]]}.

5.1. F contains the suﬃxes of all nonlinear edges.

Proposition 5.5. If E is a nonlinear edge of Gl with suﬃx u, then u is l contained in Gr(l).

l l Proof. To simplify notation we write f,G and Gr for fl,G and Gr(l). Suppose that u is not contained in Gr.IfE is crossed by the suﬃx u of an edge E then E is not a linear edge and u is not contained in Gr.We may therefore assume that E is the edge in the highest stratum. Since Gr is k f-invariant and f ﬁxes all vertices, [f (u)] is not contained in Gr for any k. Thus, the eigenray R = E · u · [f(u)] · [f 2(u)] · ... is not carried by F. The edge E determines a splitting of Fn as either a free product or an HNN-extension. Let FE denote the resulting free factor system. Since E is not an edge of Gr, F FE. Also, F = FE since FE carries R, but F does not. The argument breaks up into two cases.

Case 1. FE carries HR. In this case, the smallest free factor system F carrying F and HR is proper (since F FE), is H-invariant (since both F and HR are), and satisﬁes F F properly (since R is not carried by F). This contradicts the choice of F.

Case 2. FE does not carry HR. In this case, we will show that H contains an element of exponential growth. There is an element of H such that, when represented as a homotopy equivalence g : G → G,[g(R)] crosses inﬁnitely many E’s. The idea is that the image of a path containing E’s under a high power of f contains long initial subpaths of R, and the image under g of a path with long initial subpaths of R contains lots of E’s. This feedback gives rise to exponential growth. We now make this more precise. Let R∗ denote an initial subpath of R with the property that [g(R∗)] = STU is the concatenation of three paths such that the lengths of S and U are at least BCC(g) and such that the number of times that T crosses E and E is at least ﬁve. Let M be the length of [g(R∗)]. Choose N so that, for all paths τ starting with E of length no more than M,[f N (τ)] starts with ER∗ (see Proposition 3.18). We claim that the element of H represented by 46 MLADEN BESTVINA, MARK FEIGHN, AND MICHAEL HANDEL

N gf has exponential growth. Indeed, since Fn and the universal cover of G are quasi-isometric, it is enough to find a circuit σ in G such that the length of [[(gfN )i(g(σ))]] grows exponentially in i. We show that σ can be taken to be any circuit containing ER∗ as a subpath. If this is the case, then [[g(σ)]] contains [g(R∗)] except that perhaps initial and terminal subpaths of length less than BCC(g) may have been lost. In particular, [[g(σ)]] crosses five E’s and E’s separated by a distance of no more than M. It follows that the highest edge splitting of [[g(σ)]] induced by initial vertices of the E’s (and terminal vertices of the E’s) contains at least two subpaths crossing E’s or E’s and with length at most M. By our choice of N,[[f N g(σ)]] contains two disjoint subpaths of the form ER∗ or its inverse. So, [[gfN g(σ)]] contains two disjoint copies of [g(R∗)] or its inverse except for a loss of initial and terminal subpaths of length less than BCC(g) and so contains at least two disjoint subpaths each crossing five E’s or E’s that are separated by a distance of no more than M. This pattern continues and the number of such subpaths containing three E’s or E’s at least doubles with each application of gfN .

Corollary 5.6. (1) If i = l mod k then Ti−1 is Oi-growing if and only if there is a linear edge E in Gl whose suffix u has positive length in Ti−1; in this case the growth is linear. ⊂ (2) If σ Gl is a circuit and lTi (σ) > 0 then there is a suffix u as in (1) such that for every N>0, [[f k(σ)]] contains [f N (u)] as a subpath for all sufficiently large k.

Proof. Proposition 5.5 implies that for any circuit σ contained in Gl there k exists a constant K such that each [[fl (σ)]] has a decomposition into at most l K subpaths, each of which is either a single edge, a path contained in Gr(l),or m of the form uj for some fixed suffix uj of fl. Up to a uniform bound, the only k m terms that contribute to Ti−1([[fl (σ)]]) are those of the form uj and these contribute m · Ti−1(uj). This proves (1). For (2), we use the notation of Definition 4.27. There is no loss in assuming that the canonical decomposition of σ is a splitting. Proposition 5.5 implies that d(E) = 0 for all nonlinear edges. Lemma 4.29 then implies that at least one of the terms in the canonical decomposition of σ is a linear edge of positive length or an exceptional path of positive length. In either case, (2) follows.

5.2. Bouncing sequences stop growing.

Proposition 5.7. Ti is eventually Oi+1-nongrowing. Proof. To simplify notation we assume that i = 0 mod k. Let U be the (finite) set of suffixes of f1 that are fixed by f1. Set K = |U|. We will show that there are at most K values of i such that Ti is O1-growing. THE TITS ALTERNATIVE FOR Out(Fn) II 47

··· O Suppose to the contrary that Ti0 ,Ti1 , ,TiK are 1-growing with i0 < i1 < ···

Sublemma 5.8. There are wi ∈H,1≤ i ≤ K, and ui ∈U,0≤ i ≤ K ≥ B M such that, given M 0, [[f1 (wi(ui))]] contains [ui−1] as a subpath for all large B.

Proof of Sublemma 5.8. By Corollary 5.6, there is uK ∈Usuch that O∞ ···O∞ (u ) > 0. Since T = T − , we may approximate TiK K iK iK 1+1 2 iK −iK−1 Ni −i ON2 ···O K K−1 T by T − w where w = for suitably chosen N . iK iK 1+1 K K 2 iK −iK−1 j In particular, we may assume that (u ) > 0. In other words, TiK−1+1wK K (w (u )) > 0. Corollary 5.6 then provides a suﬃx u − , such that TiK−1+1 K K K 1 (u − ) > 0 and such that [[f B(w (u ))]] contains [uM ] as a subpath TiK−1 K 1 1 K K K−1 for all large B. The argument may be repeated starting with uK−1, etc. The sublemma follows.

We now continue with the proof of Proposition 5.7. Two of the ui’s produced in the sublemma are equal, say u0 = uK . We shall show that there exists an element in H of exponential growth, a contradiction that will establish the proposition. Let C be as in Lemma 3.17 for the UR f1, and let A be such that the length A in G of [wi(ui) ] is larger than twice BCC(wi) (with wi realized as homotopy B C+2+A equivalence on G). Choose B so that the circuit [[f wi(ui)]] contains ui−1 as a subpath. OB ···OB We claim that 1 w1 1 wK has exponential growth. Indeed, we will C+2+A show that if σ is any path in G containing L occurrences of ui whose B C+2+A interiors are disjoint, then [f1 wi(σ)] contains 2L occurrences of ui−1 with disjoint interiors. After all, when we apply wi to σ, we obtain for each oc- C+2+A C+2 A currence of ui an occurrence of [wi(ui )] (at most [wi(ui )] is lost by B our choice of A). After applying f1 , by Lemma 3.17 we see L occurrences of B 2 B 2 [f1 wi(ui )] with disjoint interiors. Finally, by our choice of B, each [f1 wi(ui )] C+2+A contains two copies of ui−1 with disjoint interiors. 5.3. Bouncing sequences never grow. Recall (Section 2.2) that, for a simplicial tree T , Arc(T ) denotes the set of conjugacy classes of stabilizers of nondegenerate arcs of T .

Lemma 5.9. Suppose that T ∈T, that O∈UPG(Fn), and that T is O-nongrowing. Set T = T O∞. Then, Arc(T ) ⊂ Arc(T ). Further, elements of Arc(T ) are O-invariant.

Proof. Let [[e]] ∈ Arc(T ). We claim that there is an arc [x, y]inT and elements a, b ∈ Fn such that 48 MLADEN BESTVINA, MARK FEIGHN, AND MICHAEL HANDEL

• [x, y] ∩ FixT (e) is nondegenerate,

• a and b are elliptic in T ,

• FixT (a) ∩ [x, y]={x}, and

• FixT (b) ∩ [x, y]={y}.

Indeed, since the axes of hyperbolic elements cover T (Theorem 2.2) there is an element c ∈ Fn such that AxisT (c) has nondegenerate overlap with the arc m FixT (e). If we choose m large enough so that c ([x, y]) is disjoint from [x, y] −m m m −m then we may take a to be c ec , b to be c ec , x to be the point in FixT (a) minimizing distance to FixT (e), and y the point in FixT (b) minimizing the distance to FixT (e). This establishes the claim. Keeping in mind that T (ab) is twice the distance in T between FixT (a) and FixT (b), we see that T (ab) >T (ae)+T (be). Since T is O-nongrowing, i eventually we have T (ab)=T (Φ (ab)), etc. where Φ ∈ Aut(Fn) represents O. Therefore, eventually Φi(a), Φi(b), and Φi(e) are elliptics in T , and i i i i T (Φ (ab)) >T (Φ (ae))+T (Φ (be)). Hence, Φ (e) is eventually a stabilizer of a nondegenerate arc of T . Since by Theorem 2.3 there are only ﬁnitely many conjugacy classes of arc stabilizers in T , it follows that the sequence {[[Φi(e)]]} takes only ﬁnitely many values, and is therefore constant by Proposition 3.16. The lemma follows.

Lemma 5.10. For all 1 ≤ l ≤ k and for all i ≥ 0,

l (1) F contains the fl-suﬃx of every edge in G .

(2) Ti is an Oi+1 nongrower.

(3) The edge stabilizers of Ti are trivial.

(4) F(Ti)=F.

Proof. The main step in the proof is to show that (2), (3) and (4) hold eventually, which is to say, for all sufficiently large i. By Proposition 5.7 there is a largest s ≥ 0 such that Ts is an Os+1 grower. By Remark 4.38 all nontrivial arc stabilizers of Ts+1 are generated by conjugates of roots of linear suffixes of fs. Lemma 5.9 implies that if e is a nontrivial edge stabilizer of Ti for i ≥ s then e is conjugate to a root of a linear suffix of fs. Lemma 5.9 also implies that the sequence {Arc(Ti)} is eventually constant and that if Arc(Ti) is not eventually trivial then there is linear suffix u of fs such that [u]isH-invariant. The free factor system given by the highest edge of Gs contains both F and [u]. Therefore, the smallest free factor system that contains both F and [u]is proper, and it is also H-invariant (since both F and [u] are), and it properly THE TITS ALTERNATIVE FOR Out(Fn) II 49 contains F (since it carries [u], while F does not). This contradicts the choice of F and shows that edge stabilizers of Ti are eventually trivial. By Proposition 4.50, {Elliptic(Ti)} eventually forms a nonincreasing sequence. Since the edge stabilizers of Ti are eventually trivial, the collection of nontrivial vertex stabilizers of Ti may be recovered from Elliptic(Ti) as the collection of maximal subgroups of Fn in the set Elliptic(Ti). So, eventually the sequence {F(Ti)} is a decreasing sequence of free factor systems. By Lemma 2.10, this sequence is eventually constant, hence eventually H-invariant and hence eventually F(Ti)=F. Having established (2) and (4) for large i, Corollary 5.6 implies (1) and then (2) for all i. Lemma 5.9 and induction then imply (3) for all i. The proof given above that (4) holds for large i now shows that (4) holds for all i.

5.4. Finding Nielsen pairs.

Definition 5.11. Let T ∈T have trivial edge stabilizers, and let H be a unipotent subgroup of Out(Fn). Assume that conjugacy classes of vertex stabilizers of T are O-invariant for all O∈H. We say that distinct nontrivial vertex stabilizers V and W of T form a Nielsen pair for H if, for all O∈H a and all lifts Φ of O to Aut(Fn) there exists a ∈ Fn such that Φ(V )=V and Φ(W )=W a. (It suffices to check this for one lift.) Here is an alternative description. Let V˜(2) denote the set of unordered pairs of distinct nontrivial vertex stabilizers of T . There is a natural diagonal (2) action of Aut(Fn)onV˜ . This action descends to an action of Out(Fn)on (2) (2) V := V˜ /Inner(Fn). If V and W are distinct nontrivial vertex stabilizers, then the corresponding element of V(2) is denoted [[V,W]]. The pair V , W is a Nielsen pair for H if [[V,W]] is a fixed point for the action of H on V(2). For example, if T is fixed by H and V , W are nontrivial stabilizers of neigh- boring vertices, then V and W form a Nielsen pair. The following Lemma 5.12 is an immediate consequence of the definition.

Lemma 5.12. Let T and H be as in Deﬁnition 5.11.

• If T is another simplicial Fn-tree that has the same vertex stabilizers as T , then two vertex stabilizers V and W form a Nielsen pair in T if and only if they form a Nielsen pair in T .

• If H = O1, O2,...,Ok and two vertex stabilizers V and W of T form a Nielsen pair for Oi for all i, then they form a Nielsen pair for H.

Proposition 5.13. Let H = O1,...,Ok be a unipotent subgroup of Out(Fn), and let T ∈T be such that

• T has trivial edge stabilizers, 50 MLADEN BESTVINA, MARK FEIGHN, AND MICHAEL HANDEL

•F(T )=F where F is maximal and H-invariant,

• T is Oi-nongrowing for all i.

Then T contains a Nielsen pair for H.

The rest of this section is devoted to the proof of Proposition 5.13, which i i appears ahead, after some preparation. By hi : G → G denote a UR for Oi such that

F F i i (1) = (Gr(i)) for some ﬁltration element Gr(i), and

i (2) if E is an edge outside Gr(i), then hi(E)=Eu for some closed path u i (depending on i and E)inGr(i).

Such a representative exists. Indeed, by Theorem 3.11 there is a UR hi satis- i fying everything except perhaps the condition that u is contained in Gr(i).It follows from Corollary 5.6 applied to H = Oi, T0 = T , and fi = hi that any hi satisfies this last condition as well. Using Lemma 5.12, we shall detect that two vertex stabilizers V and W of T form a Nielsen pair for H by examining for every i whether they form a i Nielsen pair for Oi in the tree Si obtained from the universal cover of G by i collapsing all edges that project to Gr(i). i Edge paths P in G are of the form ν0H1ν1H2 ···Hpνp where each Hj is an i i edge not in Gr(i) and each νj is a path in Gr(i). We call the subpaths νj vertical elements and the letter H is chosen for horizontal (with Gi as a model for the tree obtained from the universal cover of Gi by collapsing edges that project i to Gr(i)). Some of the νj’s could be trivial paths. When P is such a path, N (N) (N) ··· (N) then the iterates hi (P ) have a similar form ν0 H1ν1 H2 Hpνp .For { (N)} each j, the sequence νj is seen to be eventually polynomial by application N of Lemmas 4.2 and 4.7 to the pieces of the splitting of hi (P ) at the endpoints of Hk where suffixes do not develop. We say that a vertical element νj is (N) inactive if νj is independent of N. Otherwise, νj is active. Of course, hi and the edge path P are implicit in these definitions. Even trivial νj’s could be active. It follows from Lemma 4.13 applied to these same pieces that ‘inactive’ is equivalent to ‘eventually inactive’. i j When i = j there is a homotopy equivalence φij : G → G given by markings. We may assume that this map sends vertices to vertices and restricts i → j to a homotopy equivalence Gr(i) Gr(j). Let C be a constant larger than the BCC of any lift of φij to universal covers. Let ν be a vertical element in a path i j P in G . We can transfer P to another G using φij and tightening. The path [φij(ν)] has length bounded above and below by a linear function in the length of ν, and then at most 2C is added or subtracted. In particular, if the length of THE TITS ALTERNATIVE FOR Out(Fn) II 51

ν is larger than some constant C0 > 2C, then ν induces a well-deﬁned vertical element in Gj. Short ν’s can disappear and new short vertical elements can appear in [φij(P )]. Choose constants C1 ≤ C2 ≤···≤C7k such that if a vertex element ν has length ≤ Ci (0 ≤ i<7k) and is transferred to some other graph, then the induced vertex element has length ≤ Ci+1. Also, ﬁx ∈ (0, 1/14k).

Lemma 5.14. For a suﬃciently large integer m>0, the following statements hold.

2(7k−i+1)m (1) Let Ni =2 , and let Ii,l be the interval − m (1 l)Ni, (1 + l)Ni

for i =1, 2,...,7k, l =1, 2,...,14k. Then Ii,1 ⊂ Ii,2 ⊂ ··· ⊂ Ii,14k and the intervals Ii,14k are pairwise disjoint for i =1, 2,...,7k. Furthermore, the intervals Ii,14K are disjoint from [0,C7k]. (2) If a vertex element ν in an edge path P in Gi is active and has length ≤ m (1 + 14k)Ni+1 (which is the right-hand endpoint of Ii+1,14k), then the (N ) hi-iterated vertex element ν i has length in Ii,1.

i (3) If a vertex element ν in an edge path P in G has length in Ip,l (l<14k), then, after transferring to Gj, ν induces a vertex element whose length belongs to Ip,l+1.

i (4) If a vertex element ν in an edge path P in G has length in Ij,l and (Ni) Ni if i>jand l<14k, then the iterated vertex element ν in hi (P ) has length in Ij,l+1.

We think of the ﬁrst index in intervals Ii,l as measuring the order of magnitude of lengths of vertex elements. The second index is present only for technical reasons: there is a slight loss when transferring from one graph to another (3), and when applying “lower magnitude maps” (4).

Proof of Lemma 5.14. To see that the right-hand endpoint of Ii+1,14k is to the left of the left-hand endpoint of Ii,14k we have to show that

(7k−i)m (7k−i+1)m (1 + 14k)22 +m < (1 − 14k)22 i.e. that (7k−i+1)m− (7k−i)m− 1+14k 2[2 2 m] > . 1 − 14k That the latter inequality holds for large m follows from the observation that the exponent of the left-hand side − 2(7k i)m(2m − 1) − m goes to inﬁnity as m →∞. 52 MLADEN BESTVINA, MARK FEIGHN, AND MICHAEL HANDEL

It follows from Theorem 3.11(5) and (6) that there are polynomials Qi and Ri with nonnegative coefficients such that whenever ν is an active vertex element in a path P in Gi, then the length of ν(N) is in the interval [N − Ri(Gi (ν)), (1 + Gi (ν))Qi(N)]. The proof now reduces to the fact that exponential functions grow faster than polynomial functions. For example, (2) follows from the inequalities − m − Ni Ri((1+14k)Ni+1) > (1 14k)Ni and m m (1+(1+14k))Ni+1Qi(Ni) < (1 + 14k)Ni . d If we assume without loss of generality that Ri(x)=x then the first inequality simplifies to N (1+14k)d i > . m+d 14k Ni+1 Again, the left-hand side amounts to 2exp with − exp =2(7k i)m(2m − m − d) and it goes to infinity as m →∞. The proof of the second inequality and of the other claims in the lemma are similar. (For (3) use the fact that there is a linear function L such that if σ is a vertex element of a path P induced by a vertex element ν of a path P , then the length of σ is bounded by L(length(ν)).)

We will argue that if there are no H-Nielsen pairs in T , then the element ON7k ···ON2 ON1 ∈H 7k 2 1 has exponential growth. 1 1 Start with a circuit P1 in G that is not contained in Gr(1) and all of whose vertex elements have length ≤ C1. This circuit is the first generation. N1 N1 2 Then apply h1 to obtain h1 (P1) and transfer this new circuit via φ12 to G . N2 The resulting circuit P2 is the second generation. Then apply h2 and transfer 3 to G to obtain the third generation circuit P3, etc. The circuit P7k whose 7k N7k generation is 7k lives in G . Then repeat this process cyclically: apply h7k 1 st and transfer to G to get a circuit P7k+1 of (7k +1) generation etc. (N ) Suppose that ν is a vertex element of some Pi.Ifν i has length ≥ C0, (N ) then ν i induces a well-defined vertex element ν in Pi+1. We say that ν gives rise to ν. We will now label some of the vertex elements of the Pi’s with positive integers. Consider maximal (finite or infinite) chains ν1,ν2, ··· of vertex elements such that νi gives rise to νi+1. In particular, there is an integer s such that νi is a vertex element of Pi+s for i ≥ 1. If the length of the chain is ≥ 7k, then label νi by the integer i. If the chain has < 7k vertex elements, we will leave all of them unlabeled. All labels > 1inPi correspond to unique labels in Pi−1.Abirth is the introduction of label 1. A death is an occurrence THE TITS ALTERNATIVE FOR Out(Fn) II 53 of a labeled vertex element that does not give rise to a vertex element in the next generation. Any labeled vertex element can be traced backwards to its birth. Traced forward, any labeled vertex element either eventually dies, or lives forever (and the corresponding label goes to infinity).

Lemma 5.15. If a vertex element ν in some Pi is not labeled, then ν is hi-inactive and its length is ≤ C7k.

Proof. The first element ν1 of a maximal chain ν1,ν2,...,νs, s<7k, must have length ≤ C1. Indeed, assume not. Say ν1 is a vertex element in Pi+1. i By the choice of P1 we must have i ≥ 1. With a transfer to G , ν1 induces a (N ) vertex element ν of length >C0.Nowν = σ i and σ gives rise to ν1, so the chain was not maximal. If all νi’s are inactive, then the claim about the length follows from the definition of constants Ci.Ifνi is the first active element of the chain, then νi+1 has length in Ii,2 by Lemma 5.14(2). With each generation the second index of the interval increases by two until 7k generations are complete (by (3) and (4)) or its length increases to some Ij,2 with j

Lemma 5.16. If two vertex elements in Pi are labeled with no labeled vertex elements between them, then either at least one of them dies in the next

Proof. If not, then the path between two such vertex elements is a Nielsen path (i.e., its lift to T connects two vertices whose stabilizers form a Nielsen pair).

Lemma 5.17. Consider the cyclically ordered set of labels in each Pi. (1) If two labels are adjacent, at least one is < 3k.

(2) If two labels have one label between them, then at least one is < 4k.

(3) If two labels have two labels between them, then at least one is < 5k.

(4) If two labels have three labels between them, then at least one is < 6k.

Proof. Let a and b be two adjacent labels in some Pi with a, b ≥ 3k and assume that i is the smallest such i. Consider the ancestors of the two labels. According to Lemma 5.16, a death must occur between the two in some Pi−s with s

Proof of Proposition 5.13. Suppose that there are no H-Nielsen pairs in T . Let C0,C1,...,C7k and be constants as explained above. Let m be an integer satisfying Lemma 5.14, and consider the labeling of vertex elements in ON7k ···ON2 ON1 ∈H paths Pi as above. The fact that 7k 2 1 grows exponentially now follows from the observation that the number of labels in Pi+k is at least equal to the number of labels in Pi multiplied by 5/4. Indeed, consider the labels in Pi that will die before reaching Pi+k. All such labels have to be ≥ 6k (since a vertex element cannot die before reaching the ripe old age of 7k). By Lemma 5.17, any two such labels have at least three labels a, b, and c between them. By Lemma 5.16, there will be at least one birth between a and b and at least one birth between b and c between generations i + 1 and i + k. Thus the number of deaths is at most a quarter of the number of labels in Pi, and the number of births is at least twice the number of deaths. The above inequality follows.

5.5. Distances between the vertices.

Lemma 5.18. Let V and W be two vertex stabilizers of T0 and let dj denote the distance between the vertices in Tj ﬁxed by V and W .IfV and W form a Nielsen pair for H, then d0 = d1 = d2 = ···.

Proof. Choose nontrivial elements v ∈ V and w ∈ W . The distance 1 between the vertices in Tj fixed by V and W equals 2 Tj (vw) and the dis- 1 tance in Tj+1 is analogously 2 Tj+1 (vw). The latter number can be computed 1 OˆN OˆN Oˆ O as 2 Tj ( j+1(v) j+1(w)) for large N, where j+1 denotes a lift of j+1 to Aut(Fn) (since Tj is Oj+1-nongrowing). This in turn equals the distance in OˆN OˆN Tj between the vertices fixed by j+1(V ) and j+1(W ). But that equals the distance between the vertices fixed by V and W since V and W form a Nielsen pair for Oj+1.

Lemma 5.19. Let Dj ⊂ R denote the set of distances between two distinct vertices in Tj with nontrivial stabilizer. Then

(1) Dj is discrete. THE TITS ALTERNATIVE FOR Out(Fn) II 55

(2) Dj ⊇ Dj+1.

(3) There are ﬁnitely many Fn-equivalence classes of paths P joining two vertices of Tj with nontrivial stabilizer and with length(P ) = min Dj.

(4) If V and W are two nontrivial vertex stabilizers of Tj such that the distance between the corresponding vertices is min Dj, then V and W form a Nielsen pair for Oj.

(5) min Dj ≤ min Dj+1, and

(6) if min Dj = min Dj+1 then any two nontrivial vertex stabilizers V and W in Tj+1 realizing the minimal distance also realize minimal distance in Tj.

Proof. (1) Every element of Dj is a real number that can be represented as a linear combination of (ﬁnitely many) edge lengths of Tj with nonnegative integer coeﬃcients. Hence Dj is discrete. 1 OˆN OˆN (2) Every element of Dj+1 has the form 2 Tj ( j+1(v) j+1(w)) (see the proof of Lemma 5.18) and hence occurs also as an element of Dj.

(3) Let P be such a path. The quotient map Tj → Tj/Fn is either injective on P or identiﬁes only the endpoints of P , hence there are only ﬁnitely many possible images of P in the quotient graph. If two such paths have the same image, then they are Fn-equivalent.

(4) Since Oj ﬁxes Tj, for any lift Oˆj ∈ Aut(Fn)ofOj we can choose an Oˆj-invariant isometry φ : Tj → Tj. By Proposition 4.48 and Lemma 4.47, φ induces the identity in the quotient graph. Therefore the immersed path P joining the vertices ﬁxed by V and W is mapped by φ to a translate of itself (we are using the fact that all interior vertices of P have trivial stabilizer). (5) is a consequence of (2).

(6) Choose a lift Oˆj+1 ∈ Aut(Fn)ofOj+1. For large N, the distance in 1 OˆN OˆN Tj+1 between the vertices ﬁxed by V and W has the form 2 Tj ( j+1(v) j+1(w)). It follows that for large N the immersed path PN joining vertices in Tj ﬁxed OˆN OˆN by j+1(V ) and j+1(W ) has length min Dj. By (4), V and W form a Nielsen pair for hj and therefore the paths PN are translates of each other and have length min Dj.

5.6. Proof of Theorem 5.1. We are now ready for the proof of Theorem 5.1 which is reformulated as follows.

Theorem 5.20. Let H = O1, O2,...,Ok be a unipotent subgroup of Out(Fn).ByF denote a maximal H-invariant proper free factor system. Let T0 ∈T have trivial edge stabilizers and satisfy F(T0)=F. Then, the 56 MLADEN BESTVINA, MARK FEIGHN, AND MICHAEL HANDEL bouncing sequence that starts with T0 is eventually constant. The stable value T is a simplicial tree with trivial edge stabilizers, a single orbit of edges and F(T )=F.

Proof. By Lemma 5.10, the sequence consists of nongrowers, the vertex stabilizers are independent of the tree in the sequence, and all edge stabilizers are trivial. By Proposition 5.13, eventually all trees contain Nielsen pairs for H. By Lemma 5.18 it follows that the numbers min Dj of Lemma 5.19 are bounded above and hence stabilized. Say min Dj+1 = min Dj+2 = ··· = min Dj+k. Let V and W be two nontrivial vertex stabilizers in Tj+k that realize min Dj+k. By Lemma 5.19, the vertex stabilizers V and W form a Nielsen pair for every Oi, and hence for H. Let P be the embedded path joining the corresponding vertices. Since P realizes the minimum distance between vertices with nontrivial stabilizer, its projection into the quotient graph is an embedding except perhaps at the endpoints (cf. the proof of Lemma 5.19(3)). If P projects onto the quotient graph, then this quotient graph has one edge and Tj+k is ﬁxed by H.IfP does not project onto the quotient graph, we obtain a contradiction by collapsing P and its translates and thus constructing an H-invariant proper free factor system strictly larger than F.

6. Proof of the main theorem

In this section we show that Theorem 5.1 implies Theorem 1.1. Recall from the introduction that for a marked graph G with a ﬁltra- tion ∅ = G0 ⊂ G1 ⊂···⊂GK = G the set of upper triangular homotopy equivalences of G up to homotopy relative to the vertices is denoted by Q.

Lemma 6.1. Q is a group under the operation induced by composition.

Proof. Since the composition of upper triangular homotopy equivalences is clearly upper triangular, it suffices to show that if f is upper triangular, then there exists an upper triangular g such that fg(Ei) and gf(Ei) are homotopic rel endpoints to Ei for 1 ≤ i ≤ K. We define g(Ei) inductively starting with g(E1)=E1. Assume that g is defined on Gi−1 and that fg(Ej) and gf(Ej) are homotopic rel endpoints to Ej for each j

Proof that Theorem 5.1 implies Theorem 1.1. The proof is by induction on n. The n = 1 case is obvious so we may assume that Theorem 1.1 holds for rank less than n. By Theorem 5.1 there is an H invariant free factor system F represented by either one free factor of rank n − 1 or two free factors whose rank adds to n. Moreover, if F is an H-invariant proper free factor system we may assume that F F . We will give the argument in the case that F = {[[F 1]], [[F 2]]} where 1 2 Fn = F ∗ F . The remaining case is analogous; details for both cases can be found in the first part of the proof of Lemma 2.3.2 of [BFH00]. The free factor system F induces free factor systems F 1 and F 2 of F 1 and F 2. By the inductive hypothesis, there are filtered marked graphs Ki with i i filtration elements realizing F and there are lifts of H|F i to Q , the group of upper triangular homotopy equivalences of Ki up to homotopy relative to vertices. Define G to be the graph obtained from the disjoint union of K1 2 1 and K by adding an edge E with initial endpoint at a vertex v1 ∈ K and 2 i terminal endpoint at a vertex v2 ∈ K . We may assume that F is identified i with π1(K ,vi). Collapsing E to a point gives a homotopy equivalence of G 1 2 to a graph whose fundamental group is naturally identified with F ∗ F = Fn and so provides a marking on G. A filtration on K1 ∪K2 is obtained by taking unions of filtration elements of F 1 and of F 2. Adding E as a final stratum produces a filtration of G in which F is realized by a filtration element. i For each O∈H, let fi ∈Q be the lift of O|F i and let Φi be an automorphism representing O whose restriction to F i agrees with the automorphism i i induced by fi under the identification of F with π1(K ,vi). Then Φ1 = icΦ2 for some c ∈ Fn. Represent c by a closed path γ based at v1 and define f : G → G to agree with fi on Ki and by f(E)=[γE]. Then f : G → G is a topological representative of O and Corollary 3.2.2 of [BFH00] implies that, up to homotopy relative to vertices, f(E)=¯u1Eu2 for some closed paths ui ⊂ Ki. In other words f represents an element in the group Q of upper triangular homotopy equivalences of G up to homotopy relative to vertices. It remains to arrange that O → f defines a homomorphism from H to Q. It is convenient to subdivide E into edges Ei with common initial endpoint at the midpoint of E and with terminal endpoint at vi.Thusf(Ei)=Eiui and i i the fundamental groups of K ∪ Ei are identified with F . The automorphism 1 2 f# of Fn induced by f preserves both F and F ; this uniquely determines both i f#|F 1 and f#|F 2 . Replacing ui with a different path vi ⊂ K replaces f#|F i with iaf#|F i where a is the nontrivial element of Fn represented by the closed i path viu¯i.IfK has rank greater than one then f#|F i = iaf#|F i and so ui is i uniquely determined by f#|F i .IfK has rank one then f#|F i is independent of the choice of ui and we always choose ui to be trivial. In either case O → f defines a homomorphism from H to Q. 58 MLADEN BESTVINA, MARK FEIGHN, AND MICHAEL HANDEL

University of Utah, Salt Lake City, UT E-mail address: [email protected] Rutgers University, Newark, Newark, NJ E-mail address: [email protected] Lehman College, The City University of New York, Bronx, NY E-mail address: [email protected]

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